Module 7
Interval estimators
Master for Business Statistics
Dane McGuckian
Topics
7.1 Interval Estimate of the Population Mean with a Known Population Standard Deviation
7.2 Sample Size Requirements for Estimating the Population Mean
7.3 Interval Estimate of the Population Mean with an Unknown Population Standard Deviation
7.4 Interval Estimate of the Population Proportion
7.5 Sample Size Requirements for Estimating the Population Proportion
7.1
Interval Estimate of the Population Mean with a Known Population Standard Deviation
Interval Estimators
Quantities like the sample mean and the sample standard deviation are called point estimators because they are single values derived from sample data that are used to estimate the value of an unknown population parameter.
The point estimators used in Statistics have some very desirable traits; however, they do not come with a measure of certainty.
In other words, there is no way to determine how close the population parameter is to a value of our point estimate. For this reason, the interval estimator was developed.
An interval estimator is a range of values derived from sample data that has a certain probability of containing the population parameter.
This probability is usually referred to as confidence, and it is the main advantage that interval estimators have over point estimators.
The confidence level for a confidence interval tells us the likelihood that a given interval will contain the target parameter we are trying to estimate.
The Meaning of “Confidence Level”
Interval estimates come with a level of confidence.
The level of confidence is specified by its confidence coefficient – it is the probability (relative frequency) that an interval estimator will enclose the target parameter when the estimator is used repeatedly a very large number of times.
The most common confidence levels are 99%, 98%, 95%, and 90%.
Example: A manufacturer takes a random sample of 40 computer chips from its production line to construct a 95% confidence interval to estimate the true average lifetime of the chip. If the manufacturer formed confidence intervals for every possible sample of 40 chips, 95% of those intervals would contain the population average.
The Meaning of “Confidence Level”
In the previous example, it is important to note that once the manufacturer has constructed a 95% confidence interval, it is no longer acceptable to state that there is a 95% chance that the interval contains the true average lifetime of the computer chip.
Prior to constructing the interval, there was a 95% chance that the random interval limits would contain the true average, but once the process of collecting the sample and constructing the interval is complete, the resulting interval either does or does not contain the true average.
Thus there is a probability of 1 or 0 that the true average is contained within the interval, not a 0.95 probability.
The interval limits are random variables because the ...
Confidence Interval ModuleOne of the key concepts of statist.docxmaxinesmith73660
Confidence Interval Module
One of the key concepts of statistics enabling statisticians to make incredibly accurate predictions is called the Central Limit Theorem. The Central Limit Theorem is defined in this way:
· For samples of a sufficiently large size, the real distribution of means is almost always approximately normal.
· The distribution of means gets closer and closer to normal as the sample size gets larger and larger, regardless of what the original variable looks like (positively or negatively skewed).
· In other words, the original variable does not have to be normally distributed.
· This is because, if we as eccentric researchers, drew an almost infinite number of random samples from a single population (such as the student body of NMSU), the means calculated from the many samples of that population will be normally distributed and the mean calculated from all of those samples would be a very close approximation to the true population mean. It is this very characteristic that makes it possible for us, using sound probability based sampling techniques, to make highly accurate statements about characteristics of a population based upon the statistics calculated on a sample drawn from that population.
· Furthermore, we can calculate a statistic known as the standard error of the mean (abbreviated s.e.) that describes the variability of the distribution of all possible sample means in the same way that we used the standard deviation to describe the variability of a single sample. We will use the standard error of the mean (s.e.) to calculate the statistic that is the topic of this module, the confidence interval.
The formula that we use to calculate the standard error of the mean is:
s.e. = s / √N – 1
where s = the standard deviation calculated from the sample; and
N = the sample size.
So the formula tells us that the standard error of the mean is equal to the
standard deviation divided by the square root of the sample size minus 1.
This is the preferred formula for practicing professionals as it accounts for errors that may be a function of the particular sample we have selected.
THE CONFIDENCE INTERVAL (CI)
The formula for the CI is a function of the sample size (N).
For samples sizes ≥ 100, the formula for the CI is:
CI = (the sample mean) + & - Z(s.e.).
Let’s look at an example to see how this formula works.
* Please use a pdf doc. “how to solve the problem”, I have provided for you under the “notes” link.
Example 1
Suppose that we conducted interviews with 140 randomly selected individuals (N = 140) in a large metropolitan area. We assured these individuals that their answers would remain confidential, and we asked them about their law-breaking behavior. Among other questions the individuals were asked to self-report the number of times per month they exceeded the speed limit. One of the objectives of the study was to estimate (make an inference about) the average nu.
Inferential statistics are often used to compare the differences between the treatment groups. Inferential statistics use measurements from the sample of subjects in the experiment to compare the treatment groups and make generalizations about the larger population of subjects.
Findings, Conclusions, & Recommendations
Report Writing
Findings
Conclusions
Recommendations
Findings
Conclusions
Recommendations
Findings
Data
Conclusions
What the data means
Recommendations
What should we do?
Types of Reports
Proposal
Feasibility
Analysis
Annual/Quarterly
Sales/Revenue
Investment
Marketing
Research
Consumer
Research
Types of Reports
Proposal
Feasibility
Analysis
Annual/Quarterly
Sales/Revenue
Investment
Marketing
Research
Consumer
Research
Report Sections
1. Title page
2. Table of contents
3. Executive summary
4. Body sections
a. Purpose
b. Scope
c. Factors
d. Conclusions
5. References (endnotes)
Report Sections
1. Title page
2. Table of contents
3. Executive summary
4. Body sections
a. Purpose
b. Scope
c. Factors
d. Conclusions
5. References (endnotes)
New Page
New Page
New Page
New Page
New Page
Title Page
1. Title
2. Author
3. Date (use due date)
4. Audience*
5. No page number
Findings
Conclusions
Recommendations
65% of employees use Facebook
during company time.
Employees are wasting time at
work.
We should establish a social
media policy.
Findings
Conclusions
Recommendations
SHA applications are down 15%.
Exploring Report Myths
Myth Truth
Reports are entirely different
from memos and letters.
Reports may be formatted as
memos or letters.
Exploring Report Myths
Myth Truth
Reports are strictly “objective”
presentations of factual data.
Report writers use their best
judgement to select data to
provide in reports.
Exploring Report Myths
Myth Truth
Reports are mere collections
of data: they should not
incorporate the writer’s
opinion.
Reports should be adapted to
the needs of the readers.
-If readers merely need numerical or
factual data, then mere numerical or
factual data should be sufficient.
Exploring Report Myths
Myth Truth
Reports are mere collections
of data: they should not
incorporate the writer’s
opinion.
Reports should be adapted to
the needs of the readers.
-If readers rely on the report writer to
interpret the data, then the report
should incorporate the writer’s best
attempt to draw conclusions and, if
appropriate, recommendations.
Exploring Report Myths
Myth Truth
A report should be structured
as a sequence of steps in
which the writer engaged in
the “discovery process” to
collect the data.
A report should be structured
according to the needs of the
readers: to learn conclusions
or to act on recommendations.
Google Report
Hilton Annual Report
Hilton Annual Report
Aramark
Report Examples
https://storage.googleapis.com/gfw-touched-accounts-pdfs/google-cloud-security-and-compliance-whitepaper.pdf
http://ir.hilton.com/~/media/Files/H/Hilton-Worldwide-IR-V3/annual-report/Hilton_2013_AR.pdf
http://ir.hilton.com/~/media/Files/H/Hilton-Worldwide-IR-V3/annual-report/1948-Annual-Report.pdf
http://www.elon.edu/docs/e-web/bft/sustainability/ARAMARK%20Trayless%20Dining%20July ...
Confidence Interval ModuleOne of the key concepts of statist.docxmaxinesmith73660
Confidence Interval Module
One of the key concepts of statistics enabling statisticians to make incredibly accurate predictions is called the Central Limit Theorem. The Central Limit Theorem is defined in this way:
· For samples of a sufficiently large size, the real distribution of means is almost always approximately normal.
· The distribution of means gets closer and closer to normal as the sample size gets larger and larger, regardless of what the original variable looks like (positively or negatively skewed).
· In other words, the original variable does not have to be normally distributed.
· This is because, if we as eccentric researchers, drew an almost infinite number of random samples from a single population (such as the student body of NMSU), the means calculated from the many samples of that population will be normally distributed and the mean calculated from all of those samples would be a very close approximation to the true population mean. It is this very characteristic that makes it possible for us, using sound probability based sampling techniques, to make highly accurate statements about characteristics of a population based upon the statistics calculated on a sample drawn from that population.
· Furthermore, we can calculate a statistic known as the standard error of the mean (abbreviated s.e.) that describes the variability of the distribution of all possible sample means in the same way that we used the standard deviation to describe the variability of a single sample. We will use the standard error of the mean (s.e.) to calculate the statistic that is the topic of this module, the confidence interval.
The formula that we use to calculate the standard error of the mean is:
s.e. = s / √N – 1
where s = the standard deviation calculated from the sample; and
N = the sample size.
So the formula tells us that the standard error of the mean is equal to the
standard deviation divided by the square root of the sample size minus 1.
This is the preferred formula for practicing professionals as it accounts for errors that may be a function of the particular sample we have selected.
THE CONFIDENCE INTERVAL (CI)
The formula for the CI is a function of the sample size (N).
For samples sizes ≥ 100, the formula for the CI is:
CI = (the sample mean) + & - Z(s.e.).
Let’s look at an example to see how this formula works.
* Please use a pdf doc. “how to solve the problem”, I have provided for you under the “notes” link.
Example 1
Suppose that we conducted interviews with 140 randomly selected individuals (N = 140) in a large metropolitan area. We assured these individuals that their answers would remain confidential, and we asked them about their law-breaking behavior. Among other questions the individuals were asked to self-report the number of times per month they exceeded the speed limit. One of the objectives of the study was to estimate (make an inference about) the average nu.
Inferential statistics are often used to compare the differences between the treatment groups. Inferential statistics use measurements from the sample of subjects in the experiment to compare the treatment groups and make generalizations about the larger population of subjects.
Findings, Conclusions, & Recommendations
Report Writing
Findings
Conclusions
Recommendations
Findings
Conclusions
Recommendations
Findings
Data
Conclusions
What the data means
Recommendations
What should we do?
Types of Reports
Proposal
Feasibility
Analysis
Annual/Quarterly
Sales/Revenue
Investment
Marketing
Research
Consumer
Research
Types of Reports
Proposal
Feasibility
Analysis
Annual/Quarterly
Sales/Revenue
Investment
Marketing
Research
Consumer
Research
Report Sections
1. Title page
2. Table of contents
3. Executive summary
4. Body sections
a. Purpose
b. Scope
c. Factors
d. Conclusions
5. References (endnotes)
Report Sections
1. Title page
2. Table of contents
3. Executive summary
4. Body sections
a. Purpose
b. Scope
c. Factors
d. Conclusions
5. References (endnotes)
New Page
New Page
New Page
New Page
New Page
Title Page
1. Title
2. Author
3. Date (use due date)
4. Audience*
5. No page number
Findings
Conclusions
Recommendations
65% of employees use Facebook
during company time.
Employees are wasting time at
work.
We should establish a social
media policy.
Findings
Conclusions
Recommendations
SHA applications are down 15%.
Exploring Report Myths
Myth Truth
Reports are entirely different
from memos and letters.
Reports may be formatted as
memos or letters.
Exploring Report Myths
Myth Truth
Reports are strictly “objective”
presentations of factual data.
Report writers use their best
judgement to select data to
provide in reports.
Exploring Report Myths
Myth Truth
Reports are mere collections
of data: they should not
incorporate the writer’s
opinion.
Reports should be adapted to
the needs of the readers.
-If readers merely need numerical or
factual data, then mere numerical or
factual data should be sufficient.
Exploring Report Myths
Myth Truth
Reports are mere collections
of data: they should not
incorporate the writer’s
opinion.
Reports should be adapted to
the needs of the readers.
-If readers rely on the report writer to
interpret the data, then the report
should incorporate the writer’s best
attempt to draw conclusions and, if
appropriate, recommendations.
Exploring Report Myths
Myth Truth
A report should be structured
as a sequence of steps in
which the writer engaged in
the “discovery process” to
collect the data.
A report should be structured
according to the needs of the
readers: to learn conclusions
or to act on recommendations.
Google Report
Hilton Annual Report
Hilton Annual Report
Aramark
Report Examples
https://storage.googleapis.com/gfw-touched-accounts-pdfs/google-cloud-security-and-compliance-whitepaper.pdf
http://ir.hilton.com/~/media/Files/H/Hilton-Worldwide-IR-V3/annual-report/Hilton_2013_AR.pdf
http://ir.hilton.com/~/media/Files/H/Hilton-Worldwide-IR-V3/annual-report/1948-Annual-Report.pdf
http://www.elon.edu/docs/e-web/bft/sustainability/ARAMARK%20Trayless%20Dining%20July ...
This is part one of the series of learning sessions designed to understand the basics of statistics used in pharmaceutical companies.
This presentation includes the following topics:
Accuracy and Precision
Tendency of data
Sampling errors and their mitigation
Confidence interval and range
T-test
Lecture 6 Point and Interval Estimation.pptxshakirRahman10
Point and Interval Estimation:
Objectives:
Apply the basics of inferential statistics in terms of point estimation.
Compute point and estimation of population means and confidence interval.
Interpret the results of point and interval estimation.
Estimation:
Estimating the value of parameter from the sample:
An aspect of inferential statistics.
Why to estimate: Population is large enough so we
can only estimate.
Types of estimation:
Point Estimation:
A specified number value (single value) that is an estimate of a population parameter. The point estimate of the population mean µ is the sample mean.
Interval Estimate:
Range of values to estimate about population parameter.
Confidence Interval Estimation:
Range of values to estimate about population parameter.
May contain the parameter or not (Degree of confidence).
Ranges between two values.
Example:
Age (in years) 4 BScN students: 20<µ < 25 or (22.5 +2.5)
FORMULA:
Point estimate (x) + Critical Value x Standard Error.
Confidence Interval is a particular interval of estimate.
Given that sample size is large, the 95% of the sample means taken from same population and same sample size will fall in + 1.96 SD of the population mean.
Three commonly used Confidence Intervals are 90%, 95% (by default) , and 99%.
Why not too small or too large confidence intervals?
Too wide: 99.9% Interval too broad
Too narrow: 80 % More uncertainty to have population mean.
The 99% of the sample means taken from same population and
same sample size will fall in + 2.575 SD of the population mean.
Interpretation:
99% probability that interval will enclose population parameter and 1% chance that it will not have population parameter.
Level of confidence: The level of certainty that the interval will have the true population mean.
Chances of Error: Chances that the interval will not cater the true parameter.
Sum of level of confidence and chances of error =100%
📺Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.1: Estimating a Population Proportion
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.1: Estimating a Population Proportion
Chapter 6 part1- Introduction to Inference-Estimating with Confidence (Introd...nszakir
Introduction to Inference, Estimating with Confidence, Inference, Statistical Confidence, Confidence Intervals, Confidence Interval for a Population Mean, Choosing the Sample Size
Confidence Intervals in the Life Sciences PresentationNamesS.docxmaxinesmith73660
Confidence Intervals in the Life Sciences Presentation
Names
Statistics for the Life Sciences STAT/167
Date
Fahad M. Gohar M.S.A.S
1
Conservation Biology of Bears
Normal Distribution
Standard normal distribution
Confidence Interval
Population Mean
Population Variance
Confidence Level
Point Estimate
Critical Value
Margin of Error
Welcome to the presentation on Confidence Intervals of Conservation Biology on Bears.
The team will define normal distribution and use an example of variables why this is important. A standard and normal distribution is discussed as well as the difference between standard and other normal distributions. Confidence interval will be defined and how it is used in Conservation Biology and Bears. We will learn how a confidence interval helps researchers estimate of population mean and population variance. The presenters defined a point estimate and try to explain how a point estimate found from a confidence interval. Confidence level is defined and a short explanation of confidence level is related to the confidence interval. Lastly, a critical value and margin of error are explained with examples from the Statdisk.
2
Normal Distribution
A normal distribution is one which has the mean, median, and mode are the same and the standard deviations are apart from the mean in the probabilities that go with the empirical rule. Not all data has the measures of central tendency, since some data sets may not have one unique value which occurs more than once. But every data set has a mean and median. The mean is only good with interval and ratio data, while the median can be used with interval, ratio and ordinal data. Mean is used when they're a lot of outliers, and median is used when there are few.
The normal distribution is continuous, and has only two parameters - mean and variance. The mean can be any positive number and variance can be any positive number (can't be negative - the mean and variance), so there are an infinite number of normal distributions. You want your data to represent the population distribution because when you make claims from the distribution of the sample you took, you want it to represent the whole entire population.
Some examples in the business world: Some industries which use normal distributions are pharmaceutical companies. They model the average blood pressure through normal distributions, and can make medicine which will help majority of the people with high blood pressure. A company can also model its average time to create something using the normal distribution. Several statistics can be calculated with the normal distribution, and hypothesis tests can be done with the normal distribution which models the average time.
Our chosen life science is BEARS. The age of the bears can be modeled by normal distributions and it is important to monitor since that tells us the average age of the bear, and can tell us a lot about the population. If the mean is high and the standard deviatio.
inferential statistics, statistical inference, language technology, interval estimation, confidence interval, standard error, confidence level, z critical value, confidence interval for proportion, confidence interval for the mean, multiplier,
Name 1. The table shows the number of days per week, x, that 100.docxgilpinleeanna
Name
1. The table shows the number of days per week, x, that 100 students use the gym at a local high school.
x
frequency
Relative
frequency
Cumulative
frequency
0
3
1
12
2
33
3
28
4
11
5
9
6
4
1. The table shows the number of days per week, x, that 100 students use the gym at a local high school.
a. Complete the table
b. Display the information as either a pie chart, a horizontal bar chart, or a vertical bar chart.
c. Determine the mean, median, minimum frequency, maximum frequency, range, Q1, Q3 and the standard deviation, Sx
d. Based on the information and chart, what can you say about the distribution.a. Complete the table
b. Display the information as either a pie chart, a horizontal bar chart, or a vertical bar chart.
c. Determine the mean, median, minimum frequency, maximum frequency, range, Q1, Q3 and the standard deviation, Sx
d. Based on the information and chart, what can you say about the distribution.
Theme one is to identify the types of cultures or models of cultures and how they work or fit within an organization
Learning Activity #1
Using your reading material create a chart that describes the type, characteristics of the culture, associated values that would be important to keep the culture alive, and kinds of organizations structures that work best for culture. Compare and contrast them in your explanation of the chart. For instance what culture might work for Joe at the new sawmill and then which one might work at Purvis' shoe company.
Theme two: How to Create, Change, and Align Culture to the Structure and Vision.
Organizational Structure
Preface:
A leader’s job is to create the direction for the company to move forward. The leader does this in steps. Here are the steps of the process:
First, the leader designs the vision and mission for the company and second, the leader must establish an organizational structure which promotes the vision, mission and empowers the employees to keep the forward movement in the organization.
In creating the structure various factors must be considered.
· First and foremost is the purpose of the company or organization. What type of structure will best accomplish that goal? Certainly a company like UPS needs a somewhat rigid structure that is set up to focus on procedure and time sensitivity. Since UPS has as its goal to get the correct parcels to the right customers in the fastest way possible, variance in procedures or ways of accomplishing the tasks would never work. A tight delineated structure is imperative.
· Along with the purpose the leader must look at the vision of the organization. Where does the leader want the organization to go? How best can the structure provide for the future? Will the vision call for expansion into other countries or simply call for product development changes? Do you plan a struct ...
Name _____________________Date ________________________ESL.docxgilpinleeanna
Name _____________________ Date ________________________
ESL 408 Remembered Event Worksheet
1) What is the most memorable, significant event in your life?
2) What important lesson(s) or applications are there from this event?
3) Complete the chart below. Add at least 5 details to each part of the storyline.
Story Element
Details
Exposition
Rising Action
Climax
Falling Action
Resloution
...
More Related Content
Similar to Module 7 Interval estimatorsMaster for Business Statistics.docx
This is part one of the series of learning sessions designed to understand the basics of statistics used in pharmaceutical companies.
This presentation includes the following topics:
Accuracy and Precision
Tendency of data
Sampling errors and their mitigation
Confidence interval and range
T-test
Lecture 6 Point and Interval Estimation.pptxshakirRahman10
Point and Interval Estimation:
Objectives:
Apply the basics of inferential statistics in terms of point estimation.
Compute point and estimation of population means and confidence interval.
Interpret the results of point and interval estimation.
Estimation:
Estimating the value of parameter from the sample:
An aspect of inferential statistics.
Why to estimate: Population is large enough so we
can only estimate.
Types of estimation:
Point Estimation:
A specified number value (single value) that is an estimate of a population parameter. The point estimate of the population mean µ is the sample mean.
Interval Estimate:
Range of values to estimate about population parameter.
Confidence Interval Estimation:
Range of values to estimate about population parameter.
May contain the parameter or not (Degree of confidence).
Ranges between two values.
Example:
Age (in years) 4 BScN students: 20<µ < 25 or (22.5 +2.5)
FORMULA:
Point estimate (x) + Critical Value x Standard Error.
Confidence Interval is a particular interval of estimate.
Given that sample size is large, the 95% of the sample means taken from same population and same sample size will fall in + 1.96 SD of the population mean.
Three commonly used Confidence Intervals are 90%, 95% (by default) , and 99%.
Why not too small or too large confidence intervals?
Too wide: 99.9% Interval too broad
Too narrow: 80 % More uncertainty to have population mean.
The 99% of the sample means taken from same population and
same sample size will fall in + 2.575 SD of the population mean.
Interpretation:
99% probability that interval will enclose population parameter and 1% chance that it will not have population parameter.
Level of confidence: The level of certainty that the interval will have the true population mean.
Chances of Error: Chances that the interval will not cater the true parameter.
Sum of level of confidence and chances of error =100%
📺Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.1: Estimating a Population Proportion
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.1: Estimating a Population Proportion
Chapter 6 part1- Introduction to Inference-Estimating with Confidence (Introd...nszakir
Introduction to Inference, Estimating with Confidence, Inference, Statistical Confidence, Confidence Intervals, Confidence Interval for a Population Mean, Choosing the Sample Size
Confidence Intervals in the Life Sciences PresentationNamesS.docxmaxinesmith73660
Confidence Intervals in the Life Sciences Presentation
Names
Statistics for the Life Sciences STAT/167
Date
Fahad M. Gohar M.S.A.S
1
Conservation Biology of Bears
Normal Distribution
Standard normal distribution
Confidence Interval
Population Mean
Population Variance
Confidence Level
Point Estimate
Critical Value
Margin of Error
Welcome to the presentation on Confidence Intervals of Conservation Biology on Bears.
The team will define normal distribution and use an example of variables why this is important. A standard and normal distribution is discussed as well as the difference between standard and other normal distributions. Confidence interval will be defined and how it is used in Conservation Biology and Bears. We will learn how a confidence interval helps researchers estimate of population mean and population variance. The presenters defined a point estimate and try to explain how a point estimate found from a confidence interval. Confidence level is defined and a short explanation of confidence level is related to the confidence interval. Lastly, a critical value and margin of error are explained with examples from the Statdisk.
2
Normal Distribution
A normal distribution is one which has the mean, median, and mode are the same and the standard deviations are apart from the mean in the probabilities that go with the empirical rule. Not all data has the measures of central tendency, since some data sets may not have one unique value which occurs more than once. But every data set has a mean and median. The mean is only good with interval and ratio data, while the median can be used with interval, ratio and ordinal data. Mean is used when they're a lot of outliers, and median is used when there are few.
The normal distribution is continuous, and has only two parameters - mean and variance. The mean can be any positive number and variance can be any positive number (can't be negative - the mean and variance), so there are an infinite number of normal distributions. You want your data to represent the population distribution because when you make claims from the distribution of the sample you took, you want it to represent the whole entire population.
Some examples in the business world: Some industries which use normal distributions are pharmaceutical companies. They model the average blood pressure through normal distributions, and can make medicine which will help majority of the people with high blood pressure. A company can also model its average time to create something using the normal distribution. Several statistics can be calculated with the normal distribution, and hypothesis tests can be done with the normal distribution which models the average time.
Our chosen life science is BEARS. The age of the bears can be modeled by normal distributions and it is important to monitor since that tells us the average age of the bear, and can tell us a lot about the population. If the mean is high and the standard deviatio.
inferential statistics, statistical inference, language technology, interval estimation, confidence interval, standard error, confidence level, z critical value, confidence interval for proportion, confidence interval for the mean, multiplier,
Name 1. The table shows the number of days per week, x, that 100.docxgilpinleeanna
Name
1. The table shows the number of days per week, x, that 100 students use the gym at a local high school.
x
frequency
Relative
frequency
Cumulative
frequency
0
3
1
12
2
33
3
28
4
11
5
9
6
4
1. The table shows the number of days per week, x, that 100 students use the gym at a local high school.
a. Complete the table
b. Display the information as either a pie chart, a horizontal bar chart, or a vertical bar chart.
c. Determine the mean, median, minimum frequency, maximum frequency, range, Q1, Q3 and the standard deviation, Sx
d. Based on the information and chart, what can you say about the distribution.a. Complete the table
b. Display the information as either a pie chart, a horizontal bar chart, or a vertical bar chart.
c. Determine the mean, median, minimum frequency, maximum frequency, range, Q1, Q3 and the standard deviation, Sx
d. Based on the information and chart, what can you say about the distribution.
Theme one is to identify the types of cultures or models of cultures and how they work or fit within an organization
Learning Activity #1
Using your reading material create a chart that describes the type, characteristics of the culture, associated values that would be important to keep the culture alive, and kinds of organizations structures that work best for culture. Compare and contrast them in your explanation of the chart. For instance what culture might work for Joe at the new sawmill and then which one might work at Purvis' shoe company.
Theme two: How to Create, Change, and Align Culture to the Structure and Vision.
Organizational Structure
Preface:
A leader’s job is to create the direction for the company to move forward. The leader does this in steps. Here are the steps of the process:
First, the leader designs the vision and mission for the company and second, the leader must establish an organizational structure which promotes the vision, mission and empowers the employees to keep the forward movement in the organization.
In creating the structure various factors must be considered.
· First and foremost is the purpose of the company or organization. What type of structure will best accomplish that goal? Certainly a company like UPS needs a somewhat rigid structure that is set up to focus on procedure and time sensitivity. Since UPS has as its goal to get the correct parcels to the right customers in the fastest way possible, variance in procedures or ways of accomplishing the tasks would never work. A tight delineated structure is imperative.
· Along with the purpose the leader must look at the vision of the organization. Where does the leader want the organization to go? How best can the structure provide for the future? Will the vision call for expansion into other countries or simply call for product development changes? Do you plan a struct ...
Name _____________________Date ________________________ESL.docxgilpinleeanna
Name _____________________ Date ________________________
ESL 408 Remembered Event Worksheet
1) What is the most memorable, significant event in your life?
2) What important lesson(s) or applications are there from this event?
3) Complete the chart below. Add at least 5 details to each part of the storyline.
Story Element
Details
Exposition
Rising Action
Climax
Falling Action
Resloution
...
Name Bijapur Fort Year 1599 Location Bijapur city.docxgilpinleeanna
Name: Bijapur Fort
Year: 1599
Location: Bijapur city in Bijapur District of the Indian state of Karnataka
The fort precinct is studded with the historical fort, palaces, mosques, tombs and
gardens.
Built by Yusuf Adil Shah, during the rule of Adil Shahidynasty.
https://en.wikipedia.org/wiki/Bijapur,_Karnataka
https://en.wikipedia.org/wiki/Bijapur_district,_Karnataka
https://en.wikipedia.org/wiki/States_and_territories_of_India
https://en.wikipedia.org/wiki/Karnataka
https://en.wikipedia.org/wiki/Adil_Shahi
Name: Adham Khan's Tomb
Year: 1561
Location : Qutub Minar, Mehrauli, Delhi,
Built for 16th-century tomb of Adham Khan, a general of the Mughal Emperor Akbar.
It consists of a domed octagonal chamber in the Lodhi Dynasty style and Sayyid
dynasty early in the 14th century.
https://en.wikipedia.org/wiki/Qutub_Minar
https://en.wikipedia.org/wiki/Mehrauli
https://en.wikipedia.org/wiki/Delhi
https://en.wikipedia.org/wiki/Adham_Khan
https://en.wikipedia.org/wiki/Mughal_Emperor
https://en.wikipedia.org/wiki/Akbar
https://en.wikipedia.org/wiki/Lodhi_Dynasty
https://en.wikipedia.org/wiki/Sayyid_dynasty
https://en.wikipedia.org/wiki/Sayyid_dynasty
These two objects are both tomb and have it’s own style form certain dynasty.
I chose these two objects is because they are both architecture and I can talk more about
how different dynasty influences the design of the architecture.s
Week 10 Assignments – XBRL
DUE DATE: Sunday midnight of Week 6, submitted in a MS Word (or Excel if
computations required) document with filename format:
Last First_Week X hwk.doc or .xls Make sure your name appears on each page of the
homework using the header function.
Homework questions:
1. Why do you think it took from 1999, when the XBRL concept was invented, until 2009
for the SEC require that public filers adopt?
2. From the PWC Webcast on XBRL, what are the differences between the “bolt-on” and
“embedded” approach to XBRL?
3. If you worked in the Finance and Accounting department of a company, how could you
use XBRL tags to help in your job? Could XBRL tagging help other functions in a
company do their jobs?
4. US public filers are required to begin tagging and reporting financial data using XBRL
beginning in 2009. From earlier in this course, they also have many major projects that
are required now or in the coming years (IFRS, Fair Value, etc.). Aside from the obvious
benefit of job creation for CPA’s and the companies which provide these
services/software ☺, what impact do you think these requirements are going to have on
companies? Will this divert attention and resources from their core business or will this
be like all other changes they go through (e.g. SOX), an intense implementation then
business as usual?
...
Name _______________________________ (Ex2 rework) CHM 33.docxgilpinleeanna
Name: _______________________________ (Ex2 rework)
CHM 3372, Winter 2016
Exam #2 Re-work
Due Wed, 3/2/16
1. Make the ketone below from 13C-labeled formaldehyde and propane. Make certain to keep
track of your labels throughout your synthesis. (27 points)
O
Name: _______________________________ (Ex2 rework)
2. (a) The reaction below can form two possible diastereomeric products. Draw the structures of
both products, and the mechanism of the formation of either one. (4 points)
O
1. LiAlH4
2. NH4Cl, H2O
(b) What characterizes a thermodynamic product of a reaction (any reaction)? What
characterizes a kinetic product of reaction? (2 points)
(c) Which product from part (a) would you expect to be the thermodynamic product? Why? (2
points)
(d) Which product would you expect to be the kinetic product? Why? (Note that this is not
necessarily the "non-thermodynamic" product.) (2 points)
(e) When this reaction is performed, regardless of what the temperature is, only one of the two
possible products is ever formed. Which one? (1 points)
(f) Why is the other diastereomer never formed? What must occur in order for it to be formed,
which will never occur with this particular reagent? Why? (3 points)
(g) Although the other diastereomer is never formed directly in this reaction, gentle heating with
aqueous acid will isomerize the initial product into the other diastereomer. Draw the mechanism
of the isomerization, and comment on why this isomerization occurs -- why one diastereomer
will react completely to form the other. (5 points)
Name: _______________________________ (Ex2 rework)
3. This page seems like it was tough on Q#3. Let’s see if you do better the second time around.
From the three alcohols shown, provide syntheses for the molecules below. For any SN2 or E2
reactions, use only non-halogen leaving groups – use a different leaving group which was
covered in Ch. 11. (12 points)
From: Make:
OH
OH
CH3 OH
O
O
CH3
O
O
O
Name: _______________________________ (Ex2 rework)
4. (a) Once again, write the oxidation state of the metal (each complex is neutral, Nickel is
Group 10; OTf is triflate, CF3SO3-), number of d electrons, and total valence electrons for the
metal in each complex, and indicate what type of reaction is occurring. (8 points)
H Ni
OTf
PPh3
Ni
OTf
PPh3H
Ni
OTf
PPh3
Ni
OTf
PPh3
Ni
OTf
PPh3
H
(b) What are the reactant(s) and product(s) of the reaction? (This time, they are not drawn for
you.) (2 points)
(c) If the ethylene molecule were deuterated completely (CD2=CD2), where would the deuterium
atoms end up in the product? Draw the structure, showing the position(s) of the deuterium
atoms. Assume the catalytic cycle has run several times already. (2 points)
Name: _______________________________ (Ex2 rework)
5. (a) I defined a conjugated system gener ...
Name 1 Should Transportation Security Officers Be A.docxgilpinleeanna
Name:
1
Should Transportation Security Officers Be Armed?
It is the opinion of this writer that Transportation Security Officers (TSOs) should not be
armed. It is my intent to illustrate that point in this paper. During my research I will weigh the
advantages and disadvantages of arming TSOs, examining each side of the argument. I will also
offer a potential solution that while costly will still prove to be less costly than arming TSOs.
What has led to this discussion? For a majority of our society it takes years and certain
events to take place in our lives for change to occur. Those events include graduating High
School/College, getting married, or having children. In a matter of only five short minutes on
the morning of November 1st, 2013, some individual’s lives changed forever. On that morning
Paul Anthony Ciancia, age 23, opened fire in Terminal 3 of the Los Angeles International
Airport (LAX). His senseless acts killed a TSO, while injuring six other individuals. The
shooting has been debated over and over again on whether it is a terrorist act or not. The
activities before, during, and after the shooting will show the acts were certainly a terrorist
attack. But more importantly could any deaths or injuries have been avoided if the TSOs were
armed? These is the question that will continue to be debated and one that will be addressed in
this paper.
Synopsis of the event that led up to this argument:
Shortly after being dropped off at the airport by his roommate, Paul Ciancia pulled out a
rifle and began opening fire. He was carrying luggage that was filled with a semiautomatic .223
caliber Smith & Wesson M&P-15 rifle, five 30-round magazines, and hundreds of additional
rounds of ammunition ("Lax shooting suspect," 2013). Walking up to the TSA checkpoint,
Ciancia pulled out a rifle and opened fire hitting TSO Gerardo Hernandez in the chest. Ciancia
Name:
2
then apparently moved into the screening area where he continued to fire striking two other
TSOs and a male citizen. According to eye witnesses, Ciancia continually asked civilians if they
were TSA officers, when they said “no” he moved on without shooting them ("Lax shooting:
Latest," 2013). Ciancia made it as far as the food court some five minutes after the first shots
were fired. He was then surrounded by LAX police officers who engaged him in a gunfight.
Shortly after the gunfight ended Ciancia was taken into custody where he had to be transported
to a nearby trauma hospital for gunshot wounds (Abdollah, 2013).
In total eight individuals had to be treated at the scene. Four victims were treated for
gunshot wounds, while the others were treated for other injuries ("6 hospitalized after," 2013).
The sole suspect Paul Ciancia was carrying a note on him that stated he “wanted to kill TSA”
and describe them as “pigs”, the note also mentioned “fiat currency” and “NWO” ("Lax shooting
...
Name Don’t ForgetDate UNIT 3 TEST(The direct.docxgilpinleeanna
Name: Don’t Forget
Date:
UNIT 3 TEST
(The directions and procedures for this test are the same as for the previous Unit test.)
Save this test on your computer, and complete the questions by marking correct answers with the “text color” function in WORD ( ) located on the “home” toolbar.Please attach your completed test to the assignment submission page.
Section I
Please identify problems of vagueness, overgenerality and ambiguity (double meaning) in the following passages. Then explain briefly how/why the passage exemplifies that problem. (Some examples may contain more than one problem.)
1. Who was Hitler? He was an Austrian.
__vague
__overgeneral
__ambiguous
Explanation:
2. The judge sanctioned the firm's criminal conduct.
__vague
__overgeneral
__ambiguous
Explanation:
3. "Turn right here!"
__vague
__overgeneral
__ambiguous
Explanation:
4. (From a Student Code of Conduct- Sexual impropriety in the dorms after 6:00 pm is forbidden.
__vague
__overgeneral
__ambiguous
Explanation:
5. Did Donald win the election? Well, he did get quite a few votes!
__vague
__overgeneral
__ambiguous
Explanation:
6. How are Henry’s finances? Oh, he’s really quite well off!
__vague
__overgeneral
__ambiguous
7. Bertha Belch, as missionary from Africa, will be speaking tonight at the Calvary Chapel. Come and hear Bertha Belch all the way from Africa.
__vague
__overgeneral
__ambiguous
Explanation:
8. Lower Slobovia can’t be a very well-run country. I mean, it’s not particularly democratic!
[Careful: Think about the various aspects of these claims before answering.]
__vague
__overgeneral
__ambiguous
Section II. Definitions
Please indicate whether the following are stipulative, persuasive, lexical or precising definitions.
9. Postmodern means a chaotic and confusing mishmash of images and references that leaves readers and viewers longing for the days of a good, well-told story.
__ stipulative
__ persuasive
__ lexical
__ précising
10. A triangle is a plane figure enclosed by 3 straight lines.
__ stipulative
__ persuasive
__ lexical
__ precising
11. An arid region, for purposes of this study, is any region that receives an average of less than 15 inches of rain per year
__ stipulative
__ persuasive
__ lexical
__ precising
14. A Blanker is someone who sends holiday cards without signatures or personalized messages
__ stipulative
__ persuasive
__ lexical
__ precising
15. Tragedy, in literary terms, means a serious drama that usually ends in disaster nd that focuses on a single character who experiences unexpected reversals in fat, often falling from a position of authority and power because of an unrecognized flaw or misguided action
__ stipulative
__ persuasive
__ lexical
__ précising
Section III. Strategies for Defining
Please indicate whether the following lexical definitions are ostensive definitions, enumerative definitions, definitions by s ...
Name Add name hereConcept Matching From Disease to Treatmen.docxgilpinleeanna
Name: Add name here
Concept Matching: From Disease to Treatment
Using your textbooks, complete the empty squares on the table below to match specific diseases with their pathology, pathophysiology and pharmacological treatment. Be sure to use appropriate medical terminology when adding information. You should review two different sources at a minimum to develop your brief synopses.
Example of completed row:
Disease
Body system
Signs/Symptoms
Pathophysiology
Treatment(s) (Pharm & Other)
Acne vulgaris
Integumentary system
Non-inflammatory comedones or inflammatory papules, pustules or modules. Symptoms can include pain, erythema and tenderness
Release of inflammatory mediators into the skin, with follicle hyperkeratinization, Propionibacterium acne colonization, and excess production of sebum
Depending on severity, topical mediations include benzyol peroxide or retinoid drugs. Hormonal drugs (such as oral contraceptives), and in some cases antibiotics may be used for severe inflammatory acne. Nonpharmacological treatments include dermabrasion or phototherapy
Disease
Body System
Signs/Symptoms
Pathophysiology
Treatment(s)
Atopic Dermatitis
Multiple Sclerosis
Squamous cell carcinoma
Osteoporosis
Osteosarcoma
Rheumatoid arthritis
Epilepsy
Psoriasis
Alzheimer’s Disease
...
Name Abdulla AlsuwaidiITA 160Uncle VanyaMan has been en.docxgilpinleeanna
Name Abdulla Alsuwaidi
I
TA 160
"Uncle Vanya"
“Man has been endowed with reason,
with the power to create, so that he can add to what he's been given.
But up to now, he hasn't been a creator, only a destroyer.
Forests keep disappearing, rivers dry up,
wild life's become extinct, the climate's ruined,
and the land grows poorer and uglier”
The play “Uncle Vanya” written by Anton Chekhov is a pearl of the classics of Russian literature. Anton Chekhov left a great legacy in a form of his plays and short stories for the classics of world literature. Without a shadow of doubt, this masterpiece, written by one of the most prominent the Russian playwrights of his time, should be read with further analysis and discussion. “Uncle Vanya” is a realist play and Chekhov tried to make its scenes as true-to-life as possible. Chekhov spent one year writing “Uncle Vanya” and introduced a number of changes between the years 1896 – 1897. The final version of his play is famous worldwide. The plot of the play narrates a heartbreaking story of how the main hero, Ivan Petrovich Voynitsky or Uncle Vanya that was a rather calm and quiet man undergoes a moral “rebirth” developing a spirit of a rebellion. Uncle Vanya, the main hero of the play, can be characterized as a bitter aging man who spent his life in toil working for his brother-in-law. Chekhov depicted the character of uncle Vanya as a misanthrope who recognized the miserable nature of other characters.
Moreover, Chekhov’s play also involves a number of other important issues that are experienced by the play’s characters. These issues include the feeling of pointless life lacking meaning, missed opportunities, and the most touching feeling of blind admiration. It should be admitted that Chekhov used to create hidden meaning in his plays to make the readers think critically not only of his work but of their lives either. Therefore, in the play, Chekhov made every character individualistic. For instance, the central character in the play, Uncle Vanya, cares about patrimony and the Serebryakov’s family’s property. Throughout the play, uncle Vanya finds himself dismissed and rejected without the right for an opinion. Chekhov also pointed out the suffering of other characters who struggle to change their lives for better. The play consists of a number of personal dramas that are interconnected.
It can be stated that Chekhov included a number of opposite lines in his play such as the choice between obedience or riot, feeling of admiration and disrespect. The following lines from the play demonstrate the feeling of disappointment and understanding the pointlessness of a situation: “”I’m mad — but people who conceal their utter lack of talent, their dullness, their complete heartlessness under the guise of the professor, the purveyor of learned magic — they aren’t mad” (Uncle Vanya). Uncle Vanya is concerned about the wasted years and the thought of how his life could look like in case he used the opportun ...
Name Add name hereHIM 2214 Module 6 Medical Record Abstractin.docxgilpinleeanna
Name: Add name here
HIM 2214 Module 6: Medical Record Abstracting
Instructions: In this medical record abstracting assignment you will first need to download and the records (history & physical, surgery consultation, operative report, pathology report and discharge summary) for a patient with digestive system problems. (Recommend reading them in the order listed).
Save your answers to the following related questions in this document and submit them for this module's assignment.
1. Define the terms diverticulosis and diverticulitis.
2. What is the pathophysiology of diverticulitis?
3. What is a hiatal hernia?
4. Describe some of the signs or symptoms a person with a hiatal hernia might have.
5. What is a pulmonary embolus?
6. What was the etiology (cause) of the pulmonary embolus for this patient?
7. What is gastritis?
8. Which problem is likely a contributor to the patient’s Type II diabetes mellitus?
9. What was the purpose of the barium enema?
10. What does the abbreviation HEENT stand for?
11. What is thrombophlebitis?
12. What is a surgical resection?
13. Define anastomosis.
14. What is ferrous gluconate and what is it used to treat?
15. What condition is the drug Darvocet used to treat?
16. What are electrolytes?
17. What is exogenous obesity?
18. Where is the femoral pulse found/taken?
19. Where is the popliteal pulse found/taken?
20. What is hepatosplenomegaly?
21. Which condition(s) is/are the drug Humulin used to treat?
22. What is an adenocarcinoma?
23. Which condition(s) is/are the drug Lanoxin used to treat?
24. What is the purpose of ordering the blood test PTT?
25. What is a colon stricture?
26. What is/are the etiologies associated with colorectal cancer?
27. What is the medical term for gallstones?
28. Which condition(s) is the drug Zantac used to treat?
29. What does the pathology report indicate about the spread of the carcinoma in this patient?
30. What is the etiology of Type II diabetes mellitus?
· Academic arguments are designed to get someone to agree with the author, who may use pathos (emotion), logos (logic and facts) and ethos (authority and expertise) to persuade.
Academic arguments are not about ranting, screaming or otherwise increasing conflict, but in fact are the opposite: They attempt to help the other person understand what the author believes to be right (opinion) based on the evidence presented (authority, logic, facts).
For your topic for your final paper, what kinds of arguments can you develop for your claim (thesis, main idea)?
Health Record Face Sheet
Record Number:
005
Age:
67
Gender:
Male
Length of Stay:
3 days
Service:
Inpatient Hospital Admission
Disposition:
Home
Discharge Summary
Patient is a 67-year-old male. He saw the doctor recently with abdominal pain and constipation. A barium enema showed diverticulosis and perhaps a stricture near the sigmoid and rectal junction. He was scoped by the doctor, who saw a stricture at that point and sa ...
Name Sophocles, AntigoneMain Characters Antigone, Cre.docxgilpinleeanna
Name:
Sophocles, Antigone
Main Characters: Antigone, Creon (the King), Ismene (Antigone’s sister), the Chorus, the Guard, Haimon (Creon’s and Euridike’s son), Euridike (Creon’s wife/Haimon’s mother), Teiresias (the prophet), the messenger.
1. Aristotle writes that the tragic hero suffers from a harmartia or error. Who is the tragic hero of the play? Why do you think so?
2. Who is in the right? Antigone? Creon? Both? Neither? Why?
3. What makes this play tragic?
4. What is the role of the chorus in this production? How do they fit into the play?
5. What do you think about the way the production differentiates between divine law and human law? Which characters do you think are more closely linked to what (kind of) law?
6. Why is this art? What is the relationship between Antigone and a painting or a statue, such that we can call them both art?
...
N4455 Nursing Leadership and ManagementWeek 3 Assignment 1.docxgilpinleeanna
N4455 Nursing Leadership and Management
Week 3 Assignment 1: Financial Management Case Study v2.2
Name:
Date:
Overview: Financial Management Case Study
One of the important duties of a nurse leader is to manage personnel and personnel budgets. In this assignment, you will assume the role of a nurse manager. You will use given data to make important decisions regarding budgets and staffing.
Some nurse managers have computer spreadsheets or software applications to help them make decisions regarding budgets and staffing. You will only need simple mathematical operations* to perform the needed calculations in this assignment because the scenario has been simplified. Furthermore, some data have been provided for you that a nurse leader might need to gather or compute in a real setting. Still, you will get a glimpse of the complexity of responsibilities nurse leaders shoulder regarding financial management.
· To calculate the percent of the whole a given number represents, follow these steps:
Change the percentage to a decimal number by moving the decimal twice to the left (or dividing by 100).
Multiply the new decimal number by the whole.
Example: What is 30% of 70?
30%= .30; (.30) × 70 = 21
· To find out what percentage a number represents in relation to the whole, follow these steps:
Divide the number by the whole (usually the small number by the large number).
Change the decimal answer to percent by moving the decimal twice to the right (or multiplying by 100).
Example: What percent of 45 is 10?
10 ÷ 45 = .222; so, 10 is 22% of 45.
* You will only need addition, subtraction, multiplication, and division.
Case Study
You are the manager for 3 West, a medical/surgical unit. You have been given the following data to assist you in preparing your budget for the upcoming fiscal year.
Patient Data
ADC: 54
Budget based on 5.4 Avg. HPPD
(5.4 HPPD excludes head nurse and unit secretaries)
Staff Data
Total FTEs
37.0 Variable FTEs
1.0 Nurse Manager
2.2 Unit Secretaries
40.2 Total FTEs
Staffing Mix
RN
65%
LVN
20%
NA
15%
Average Salary Scale per Employee
(Fringe benefits are 35% of salaries)
Nurse Manager
$77,999.00 per year
Registered Nurses (RN)
$36.00 per hour
Licensed Vocational Nurses (LVN)
$24.00 per hour
Nurse Aides (NA)
$13.50 per hour
Unit Secretary (US)
$11.25 per hourRubric
Use this rubric to guide your work on this assignment.
Criteria
Target
Acceptable
Unacceptable
Question 1
Both % and FTEs column totals within ± 2 of correct answers
(13-16 Points)
Either % or FTEs column totals within ± 2 of correct answers
(5-12 points)
Neither % nor FTEs column totals within ± 2 of correct answers
(0-4 points)
Question 2
All column (except Hours and Salary) totals within ± 2 of correct answers
(17-20 Points)
At least 4 column totals within ± 2 of correct answers
(5-16 points)
Less than 4 column totals within ± 2 of correct answers
(0-4 points)
Question 3
A. Table
All ...
Name Habitable Zones – Student GuideExercisesPlease r.docxgilpinleeanna
Name:
Habitable Zones – Student Guide
Exercises
Please read through the background pages entitled Life, Circumstellar Habitable Zones, and The Galactic Habitable Zone before working on the exercises using simulations below.
Circumstellar Zones
Open the Circumstellar Zone Simulator. There are four main panels:
· The top panel simulation displays a visualization of a star and its planets looking down onto the plane of the solar system. The habitable zone is displayed for the particular star being simulated. One can click and drag either toward the star or away from it to change the scale being displayed.
· The General Settings panel provides two options for creating standards of reference in the top panel.
· The Star and Planets Setting and Properties panel allows one to display our own star system, several known star systems, or create your own star-planet combinations in the none-selected mode.
· The Timeline and Simulation Controls allows one to demonstrate the time evolution of the star system being displayed.
The simulation begins with our Sun being displayed as it was when it formed and a terrestrial planet at the position of Earth. One can change the planet’s distance from the Sun either by dragging it or using the planet distance slider.
Note that the appearance of the planet changes depending upon its location. It appears quite earth-like when inside the circumstellar habitable zone (hereafter CHZ). However, when it is dragged inside of the CHZ it becomes “desert-like” while outside it appears “frozen”.
Question 1: Drag the planet to the inner boundary of the CHZ and note this distance from the Sun. Then drag it to the outer boundary and note this value. Lastly, take the difference of these two figures to calculate the “width” of the sun’s primordial CHZ.
CHZ Inner Boundary
CHZ Outer Boundary
Width of CHZ
NAAP – Habitable Zones 1/7
Question 2: Let’s explore the width of the CHZ for other stars. Complete the table below for stars with a variety of masses.
Star Mass (M )
Star Luminosity (L )
CHZ Inner Boundary (AU)
CHZ Outer Boundary (AU)
Width of CHZ (AU)
0.3
0.7
1.0
2.0
4.0
8.0
15.0
Question 3: Using the table above, what general conclusion can be made regarding the location of the CHZ for different types of stars?
Question 4: Using the table above, what general conclusion can be made regarding the width of the CHZ for different types of stars?
Exploring Other Systems
Begin by selecting the system 51 Pegasi. This was the first planet discovered around a star using the radial velocity technique. This technique detects systematic shifts in the wavelengths of absorption lines in the star’s spectra over time due to the motion of the star around the star-planet center of mass. The planet orbiting 51 Pegasi has a mass of at least half Jupiter’s mass.
Question 5: Zoom out so that you can compare this planet to those in our solar system (you can click-hold-drag to change t ...
Name Class Date SKILL ACTIVITY Giving an Eff.docxgilpinleeanna
Name Class Date
SKILL ACTIVITY
Giving an Effective Presentation
Directions: Read the information about oral presentations. Then
complete an outline for your own presentation.
One kind of oral presentation is a speech in which you explain
a position, or opinion, about an issue. After your speech, the
audience asks questions and you answer them. Preparing is the
first step. Use the following list as a guide to prepare.
• Decide what opinion you will take—for or against—and why.
• Write a short opening statement that gives your opinion.
• Gather facts and examples that support your opinion.
• Write a short conclusion that restates your opinion.
• Brainstorm a list of questions that your audience might ask.
Write down answers to the questions.
• Practice your presentation. Keep track of how long your
speech takes.
When you make the presentation, follow these steps:
• Begin with your opening statement.
• Give facts and examples that support your opinion.
• Conclude by stating your opinion again in different words.
• Answer questions from the audience. Listen carefully to make
sure you understand each question.
• While you are speaking, remember to look at your audience.
• Speak loudly and clearly so they can hear you.
Directions: Prepare and give a presentation on the following
topic: Is the increase in temporary employment a good thing for
American workers? Copy the following outline onto your own
paper to begin organizing your ideas.
I. Your opening statement:
II. Facts and examples that support your opinion:
1–5.
III. Your conclusion:
IV. Questions the audience may ask:
1–5.
V. Answers to these questions:
1–5.
BODY%RITUAL%AMONG%THE%NACIREMA%%
Horace%Miner%
%
From%Horace%Miner,%"Body%Ritual%among%the%Nacirema."%Reproduced%by%permission%of%the%
American%Anthropological%Association%from%The%American%Anthropologist,%vol.%58%(1956),%pp.%
503S507.%
%
Most%cultures%exhibit%a%particular%configuration%or%style.%A%single%value%or%pattern%of%perceiving%
the%world%often%leaves%its%stamp%on%several%institutions%in%the%society.%Examples%are%"machismo"%
in%Spanish>influenced%cultures,%"face"%in%Japanese%culture,%and%"pollution%by%females"%in%some%
highland%New%Guinea%cultures.%Here%Horace%Miner%demonstrates%that%"attitudes%about%the%
body"%have%a%pervasive%influence%on%many%institutions%in%Nacireman%society.%
The%anthropologist%has%become%so%familiar%with%the%diversity%of%ways%in%which%different%peoples%
behave%in%similar%situations%that%he%is%not%apt%to%be%surprised%by%even%the%most%exotic%customs.%
In%fact,%if%all%of%the%logically%possible%combinations%of%behavior%have%not%been%found%somewhere%
in%the%world,%he%is%apt%to%suspect%that%they%must%be%present%in%some%yet%undescribed%tribe.%%This%
point%has,%in%fact,%been%expressed%with%respect%to%clan%organization%by%Murdock.%In%this%light,%
the%magical%beliefs%and%practices%of%the%Nacirema%present%such%unusual%aspect ...
Name Speech Title I. Intro A) Atten.docxgilpinleeanna
Name:
Speech Title
I. Intro:
A) Attention getter --
B) Purpose Statement --
C) Thesis --
II. BODY
A) Main Point Number 1:
a)
b)
c)
transition --
B) Main Point Number 2:
a)
b)
c)
transition --
C) Main Point Number 3:
a)
b)
c)
transition –
III. CONCLUSION:
A) Summary statement --
B) Memorable conclusion --
References
List all references on a separate page with the word “References” centered at the top.
Name: Suepin Nguyen
Hygiene Saves Lives
I. Intro: To give an informational speech about Ignaz Philipp Semmelweis
A) Attention getter – On each square centimeter of your skin, there are about 1,500
bacteria. That’s a lot of germs. According to a study conducted by Michigan State
University researchers, 95% of people do not properly wash their hands long enough to
kill the infection causing germs and bacteria (Jaslow, “95 Percent of People Wash Their
Hands Improperly: Are You One of Them?”).
B) Purpose Statement - That’s gross. While I can’t force you to wash your hands, perhaps
today I can help you realize just how much history and evidence is behind this crucial
bathroom ritual.
C) Thesis – Today, I will inform you all about Ignaz Philipp Semmelweis by discussing first
about his practice and studies, second about his scientific methods that saved a lot of
lives, and third about the germ theory we all take for granted.
II. BODY:
A) Main Point Number 1: To begin, I want to introduce Ignaz Philipp Semmelweis.
a) Ignaz Semmelweis became a physician and earned his doctorate degree in medicine
in 1844. This time period was known as the start of the golden age of the physician
scientist” (NPR.org). This means that doctors were expected to have scientific
training. Doctors were more interested in numbers and collecting data (Justin Lessler,
an assistant professor at Johns Hopkins School of Public Health).
b) In 1846, Dr. Semmelweis showed up for his new job in the maternity clinic at the
General Hospital in Vienna. Due to the time period, Dr. Semmelweis thought like a
physician scientist and wanted to figure out why so many women in maternity wards
were dying from childbed fever (Davis, “The Doctor Who Championed
Hand-Washing and Briefly Saved Lives”).
c) So what did he do? He collected data of his own. He studied two maternity wards in
the hospital. One was staffed by all male doctors and medical students, and the other
by female midwives. He tallied up the number of deaths in each ward and found that
women in the clinic staffed by doctors and medical students died at a rate 5 times ...
n engl j med 352;16www.nejm.org april 21, .docxgilpinleeanna
n engl j med
352;16
www.nejm.org april
21, 2005
1630
P E R S P E C T I V E
verse populations and less inclusive health care pro-
grams, cautioned Joanne Lynn, a senior research-
er with the RAND Corporation and director of the
Washington Home Center for Palliative Care Stud-
ies in Washington, D.C. “There isn’t a huge demand
for assisted suicide in good care systems, but there
could be a huge demand in much less adequate care
systems,” Lynn said.
Psychiatrist Linda Ganzini of Oregon Health and
Sciences University agrees that her state’s high-
quality system of palliative care is the factor most
responsible for keeping the number of assisted-sui-
cide cases low. “Your safety net is your end-of-life
care and your hospice care,” she said. “It’s not the
safeguards that you build into the law.”
1.
Colburn D. Why am I not dead? The Oregonian. March 4,
2005:A01.
2.
Tolle SW, Tilden VR, Drach LL, Fromme EK, Perrin NA, Hedberg
K. Characteristics and proportion of dying Oregonians who person-
ally consider physician-assisted suicide. J Clin Ethics 2004;15:111-8.
3.
Ganzini L, Nelson HD, Lee MA, Kraemer DF, Schmidt TA,
Delorit MA. Oregon physicians’ attitudes about and experiences
with end-of-life care since passage of the Oregon Death with Dig-
nity Act. JAMA 2001;285:2363-9.
4.
House of Lords Select Committee on the Assisted Dying for
the Terminally Ill Bill. Volume I: Report. HL Paper 86-I.
The story of Terri Schiavo should be disturbing to
all of us. How can it be that medicine, ethics, law,
and family could work so poorly together in meet-
ing the needs of this woman who was left in a per-
sistent vegetative state after having a cardiac ar-
rest? Ms. Schiavo had been sustained by artificial
hydration and nutrition through a feeding tube
for 15 years, and her husband, Michael Schiavo, was
locked in a very public legal struggle with her par-
ents and siblings about whether such treatment
should be continued or stopped. Distortion by inter-
est groups, media hyperbole, and manipulative use
of videotape characterized this case and demon-
strate what can happen when a patient becomes
more a precedent-setting symbol than a unique hu-
man being.
Let us begin with some medical facts. On Feb-
ruary 25, 1990, Terri Schiavo had a cardiac arrest,
triggered by extreme hypokalemia brought on by an
eating disorder. As a result, severe hypoxic–ische-
mic encephalopathy developed, and during the sub-
sequent months, she exhibited no evidence of high-
er cortical function. Computed tomographic scans
of her brain eventually showed severe atrophy of
her cerebral hemispheres, and her electroenceph-
alograms were flat, indicating no functional activ-
ity of the cerebral cortex. Her neurologic examina-
tions were indicative of a persistent vegetative state,
which includes periods of wakefulness alternating
with sleep, some reflexive responses to light and
noise, and some basic gag and swallowing respons-
es, but no signs of emotion, wi ...
Name:
Class:
Date:
HUMR 211 Spring 2018 - Midterm
Copyright Cengage Learning. Powered by Cognero. Page 1
Indicate the answer choice that best completes the statement or answers the question.
1. Each of the following is considered the business of social welfare except:
a. telling people how to live their lives.
b. ending all types of discrimination and oppression.
c. providing child-care services for parents who work outside the home.
d. rehabilitating people who are addicted to alcohol or drugs.
2. Which of the following statements is consistent with the residual view of social welfare?
a. Recipients are viewed as being entitled to social services and financial help.
b. Social services and financial help should be provided to an individual on a short-term basis, primarily during
emergencies.
c. It is associated with the belief that an individual’s difficulties are due to causes largely beyond his or her
control.
d. There is no stigma attached to receiving funds or services. In this view, when difficulties arise, causes are
sought in the society, and efforts are focused on improving the social institutions within which the individual
functions.
3. Which of the following is consistent with an institutional view of social welfare?
a. Social services and financial aid should be provided only when other measures or efforts have been exhausted.
b. Causes for client’s difficulties are sought in the society.
c. Clients are to blame for their predicaments because of personal inadequacies.
d. Recipients are required to perform certain low-grade work assignments to receive financial aid.
4. The Elizabethan Poor Law of 1601 established three categories of relief recipients:
a. the insane, the poor, and the disabled.
b. the insane, dependent children, and the poor.
c. the able-bodied poor, the impotent poor, and dependent children.
d. the disabled, wives of prisoners, and the poor.
5. Before 1930 social services and financial assistance for people in need were provided primarily by _____.
a. churches and voluntary organizations
b. federal and state institutions
c. richer European countries
d. the military
6. President Clinton and the Republican-controlled Congress abolished Aid to Families with Dependent Children (AFDC)
in 1996 and replaced it with:
a. Welfare Services for Single Mothers.
b. Temporary Assistance to Needy Families.
c. Conditional Aid to Single Parents.
d. Assistance for Poor Families.
Indicate whether the statement is true or false.
Name:
Class:
Date:
HUMR 211 Spring 2018 - Midterm
Copyright Cengage Learning. Powered by Cognero. Page 2
7. One of the businesses of social welfare is to provide adequate housing for the homeless.
a. True
b. False
8. In the past, social welfare has been more of a pure sci ...
NAME ----------------------------------- CLASS -------------- .docxgilpinleeanna
NAME ----------------------------------- CLASS -------------- DATE -----------
THE
Source Articles from
WALL STREET JOURNAL.
CLASSROOM EDITION
Chapter 17 International Trade
This article from the April2004 Wall Street Journal Classroom Edition offers a
broader view of a long-running trend in global trade: the movement of manufacturing
jobs to other countries. In "Two-Way Street," Journal staff reporters Joel Millman
and Norihiko Shirouzu explain that while many manufacturing jobs are indeed
streaming out of the U.S., some foreign companies are eagerly creating new manufac-
turing jobs in the U.S.
Before reading the article, you may want ro look up the following terms: proxim-
ity, incentives, rhetoric, value chain.
uBut free trade works both
ways, and just as U.S.
companies look overseas
for workers, a lot of foreign
companies have been
expanding their operations
in the U.S. and creating new
jobs for Americans. The
attractions for them are better
business conditions, proxim-
ity to the ever-expanding U.S.
consumer market, and the
promise of incentives that
many U.S. communities offer
to attract new investment.''
Free trade has hammered a lot of U.S. towns, making it easier for companies to send manu-facturing jobs south of
the border or overseas, and
idling hundreds of American
factories and tens of thousands
of workers.
But free trade works both
ways, and just as U.S. compa-
nies look overseas for workers,
a lot of foreign companies have
been expanding their opera-
tions in the U.S. and creating
new jobs for Americans. The
attractions for them are better
business conditions, proximity
to the ever-expanding U.S.
consumer market, and the
promise of incentives that
many U.S. communities offer
to attract new investment.
In 1999, for example,
Gruma, Mexico's largest pro-
ducer of corn flour and tor-
tillas, wanted to extend its
sa les territory in the eastern
U.S. The manufacturer found
that the quickest way was to
buy a rival, Barnes Foods, ven-
dor of the regional Pepito
brand in Goldsboro, N .C ..
After closing the $12 million
transaction, Gruma found something else: a com-
munity eager to offer incentives to persuade the
Mexican company to invest
millions more.
Within a year, Gruma
delighted Goldsboro by agree-
ing to buy an empty warehouse
the city owned outside rown.
The building had sat for four
years, after officials spent more
than $1 million trying to mar-
ket it as parr of an industrial
park. By promising to invest
$13 million locally, and add
100 jobs to Barnes's payroll,
Gruma got $200,000 chopped
off the building's sale price and
another $200,000 in grants to
defray infrastructure costs.
Gruma also received job-cre-
ation tax credits to offset
almost $200,000 annually
from its state corporate income
tax. Ultimately, the Mexican
company well exceeded the
n urn ber of new hires it
promised, tripling its Golds-
boro work force to nearly 200. ...
Name Understanding by Design (UbD) TemplateStage 1—Desir.docxgilpinleeanna
Name:
Understanding by Design (UbD) Template
Stage 1—Desired Results
Q Established Goals:
Students will understand to add and subtract of the numbers.
Understandings:
The student will understand some of the terms and symbols that are very important to add or subtract numbers.
Essential Questions:
What does the mean plus or add?
How can we find the different between two numbers?
What does “=” mean? And when can we use it?
Students will know the most popular of the three symbols:
1- "+" to add the numbers.
2- "-" to subtract the numbers.
3- "=" to equal the numbers.
Students will be able to
· Use the terms 'add, plus, equals, minus, and the difference between them'.
· Use number line to model and determine the difference between two numbers, e.g. “Difference between 7 and 4 is 3”.
· Use the symbols for plus (+), minus (–) and equals (=).
Stage 2 – Assessment Evidence
Performance Tasks:
•
I am math teacher (R) and I have been hired by the principal and council (A) of The School of Riyadh for elementary students who are 11 years old. The exercise will target the addition and subtraction of the math. I must illustrate and define each one. (G) I am going to use audio aids in teaching them then the students are going to write down new ideas in a table that I have made in a booklet. (P) The table contains topics and underneath each topic, there are three boxes that contain each pillar. Each box has to have the particular picture that defines each pillar and must be colorful as well. All this is in stapled booklet. (S) The cost of the tablet and booklets are $100. This task must be completed in one week (S2).
Other Evidence:
How were addition and a subtraction derived? (E)
How is addition translated to sunbathing? (I)
How is addition or subtraction use in our world? (A)
How does addition compare to subtraction? (P)
How can I best recognize addition and subtraction? (SK)
Stage 3 – Learning Plan
Learning Activities:
Differentiated Instruction
White Cube
(Basic Level)
EXPLAIN
Big Idea:
INTERPRET
APPLICATION
PERSPECTIVE
Unit:
Cubing Examples
using
the Six
Facets of
Understanding
EMPATHY
SELF-KNOWLEDGE
Differentiated Instruction
Red Cube
(Intermediate Level)
KNOWLEDGE
Big Idea:
.
COMPREHENSION
APPLICATION
ANALYSIS
Unit:
Cubing Examples
using
Bloom’s Taxonomy
SYNTHESIS
EVALUATION
Differentiated Instruction
(Advanced Level)
ThinkDOTS
Sternberg’s Triarchic Model
●
SC
Big Idea:
●●
SA
●●●
SC
●●
●●
SP
Unit:
Cubing Examples
using
ThinkDots and
Sternberg
●●●
●●
SP
KEY:
SC = Creativity
SP = Practical
SA = Analytical
●●●
●●.
SA
G
U
Q
S
T�
OE
L
L
Running head: KEEPING SCORE 1
Keeping Score
Jillian Grantham
Grantham University
KEEPING SCORE 2
Abstract
Proposed changes to Little League scoring policies can seriously affect the elements that make
this game not only popular, but beneficial to th ...
Name MUS108 Music Cultures of the World .docxgilpinleeanna
Name MUS108 Music Cultures of the World Points /40
Winter 2018 Exam 2
(Take Home, open notes – NOT open book)
Matching – (1 point each, 8 points total)
Match each term with one of the following cultures by writing the corresponding letter in the blank space:
A. India
B. Bali
C. Ireland
1. _______sitar
2._______kilitan telu
3._______kecak
4._______gamelan
5._______Sean-nós
6._______beleganjur
7._______alap
8._______céilí
9. Describe Irish music. Please include information from each of the 3 different “eras” discussed in the book. (4 points)
10. Describe a raga in detail, with much attention paid to form, instruments, and development/barhat. (4 points)
11. What effect did the potato famine have on the culture and music of Ireland? (6 points)
12. What is ombak? Please explain it in detail, including how it is achieved. (4 points)
13. What is the difference between ceili and session? (2 points)
5. Listening Exercise – 12 points ( 4 points each) Sound Files are on Moodle!!!
Listen to the sound clips. See if you can guess what culture/tradition they come from. You may even be able to guess the type/form of music. Please write down your thought process. What are the clues? Why might it be from one particular culture? Listen to instruments, form, texture. The right answer is not the goal. What I need to see is your reasoning. You could get full credit even if you guess the wrong culture, provided your reasoning is sound. Complete sentences are not needed; lists are fine.
Clip 1.
Clip 2.
Clip 3.
...
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
Delivering Micro-Credentials in Technical and Vocational Education and TrainingAG2 Design
Explore how micro-credentials are transforming Technical and Vocational Education and Training (TVET) with this comprehensive slide deck. Discover what micro-credentials are, their importance in TVET, the advantages they offer, and the insights from industry experts. Additionally, learn about the top software applications available for creating and managing micro-credentials. This presentation also includes valuable resources and a discussion on the future of these specialised certifications.
For more detailed information on delivering micro-credentials in TVET, visit this https://tvettrainer.com/delivering-micro-credentials-in-tvet/
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
MATATAG CURRICULUM: ASSESSING THE READINESS OF ELEM. PUBLIC SCHOOL TEACHERS I...NelTorrente
In this research, it concludes that while the readiness of teachers in Caloocan City to implement the MATATAG Curriculum is generally positive, targeted efforts in professional development, resource distribution, support networks, and comprehensive preparation can address the existing gaps and ensure successful curriculum implementation.
Module 7 Interval estimatorsMaster for Business Statistics.docx
1. Module 7
Interval estimators
Master for Business Statistics
Dane McGuckian
Topics
7.1 Interval Estimate of the Population Mean with a Known
Population Standard Deviation
7.2 Sample Size Requirements for Estimating the Population
Mean
7.3 Interval Estimate of the Population Mean with an Unknown
Population Standard Deviation
7.4 Interval Estimate of the Population Proportion
7.5 Sample Size Requirements for Estimating the Population
Proportion
7.1
Interval Estimate of the Population Mean with a Known
Population Standard Deviation
Interval Estimators
Quantities like the sample mean and the sample standard
2. deviation are called point estimators because they are single
values derived from sample data that are used to estimate the
value of an unknown population parameter.
The point estimators used in Statistics have some very desirable
traits; however, they do not come with a measure of certainty.
In other words, there is no way to determine how close the
population parameter is to a value of our point estimate. For
this reason, the interval estimator was developed.
An interval estimator is a range of values derived from sample
data that has a certain probability of containing the population
parameter.
This probability is usually referred to as confidence, and it is
the main advantage that interval estimators have over point
estimators.
The confidence level for a confidence interval tells us the
likelihood that a given interval will contain the target parameter
we are trying to estimate.
The Meaning of “Confidence Level”
Interval estimates come with a level of confidence.
The level of confidence is specified by its confidence
coefficient – it is the probability (relative frequency) that an
interval estimator will enclose the target parameter when the
estimator is used repeatedly a very large number of times.
The most common confidence levels are 99%, 98%, 95%, and
90%.
Example: A manufacturer takes a random sample of 40
computer chips from its production line to construct a 95%
confidence interval to estimate the true average lifetime of the
chip. If the manufacturer formed confidence intervals for every
possible sample of 40 chips, 95% of those intervals would
contain the population average.
3. The Meaning of “Confidence Level”
In the previous example, it is important to note that once the
manufacturer has constructed a 95% confidence interval, it is no
longer acceptable to state that there is a 95% chance that the
interval contains the true average lifetime of the computer chip.
Prior to constructing the interval, there was a 95% chance that
the random interval limits would contain the true average, but
once the process of collecting the sample and constructing the
interval is complete, the resulting interval either does or does
not contain the true average.
Thus there is a probability of 1 or 0 that the true average is
contained within the interval, not a 0.95 probability.
The interval limits are random variables because their values
depend upon the results of a random sample of data.
However, once they are calculated from a particular sample,
those limits are no longer random variables – they become fixed
constants, so speaking about their probability in terms of the
confidence level is no longer valid.
The Meaning of “Confidence Level”
In the diagram, there are 8 different
confidence intervals represented. Each
confidence interval was constructed using
a sample of size n, drawn from the same
population, and all of the intervals have a
95% level of confidence. The vertical line
in the diagram indicates where the popula-
tion mean is located.
7 of the intervals capture the population mean, but the second
interval does not.
4. If we looked as a very large number of these intervals,
approximately 5% (100% - 95%) of them would fail to include
the mean.
All of the others would contain the mean as expected.
Confidence Interval for Estimating a Population Mean (with
Sigma Known)
The formula for the confidence interval to estimate the mean
consists of two values; a lower limit and an upper limit.
Confidence Interval for
where n = the sample size
σ = the population standard deviation
= the sample mean
= the z-score separating an area of in the upper tail of the
standard normal curve
The formula is often expressed as: Lower LimitUpper Limit
Confidence Interval for Estimating a Population Mean (with
Sigma Known)
However, the most common way to express the formula for the
confidence interval to estimate the mean is
5. where is called the margin of error.
The Margin of Error
In the formula for the confidence interval to estimate the
population mean (with σ known), there is a quantity called the
margin of error.
The margin of error is the maximum likely difference observed
between the sample mean and the population mean , and it is
denoted by E.
The margin of error for the confidence interval to estimate the
mean is given by the following formula:
where n = the sample size
σ = the population standard deviation
= the sample mean
= the z-score separating an area of in the upper tail of the
standard normal curve
The Margin of Error
The margin of error is what determines the width of the
confidence interval.
The width of a confidence interval is given by:
When estimating the mean using a confidence interval, the
smaller the margin of error, the better.
Since the confidence interval is designed to contain the mean, a
narrow interval gives us a better idea of where the mean is
located.
6. The Importance of a Known Population Standard Deviation for
a Confidence Interval
The population standard deviation for a random variable is part
of the margin of error formula used to estimate the population
mean of that random variable.
The reason for this is that the population standard deviation is
needed to determine the precise standard error of the sample
mean.
If we do not know the precise standard error of the sample
mean, we cannot guarantee the level of confidence specified for
the interval.
The standard error for :
The margin of error used to estimate
The critical value is determined by assuming the distribution of
the sample mean is normally distributed with an unknown mean
of and a known standard error of .
The Importance of a Known Population Standard Deviation for
a Confidence Interval
If we do not know , then we cannot know the standard error of
the sample mean.
This would prevent us from stating an accurate confidence level
for our interval estimate.
For this reason, we should know the population standard
deviation when using the following confidence interval formula
to estimate :
Formula to estimate (with known):
7. How to Find Critical Z Values
Find the critical Z value needed to construct a 90% confidence
interval: .
From the normal table,
looking up the probability
value of 0.4500, we get
because 0.4500 fell
between 1.64 and 1.65.
The Margin of Error when Sigma is Known
The mean quarterly earnings per share for a sample of 36 stocks
is $10.52, and the population standard deviation is $2.50.
Calculate the margin of error that would be used when
estimating the population mean with a confidence level of 95%.
The margin of error is given by:
Here n = 36, , CL = 0.95, So
Then
8. The Steps to Create a Confidence Interval when Sigma is
Known
Step 1: Gather the sample size, sample mean, population
standard deviation, and a confidence level
Step 2: Find the Z critical value
Step 3: Calculate the margin of error
Step 4: Calculate the confidence interval
The Confidence Interval for the Mean when Sigma is Known
A student wants to estimate the average amount of time it takes
to commute to campus from her apartment. For a random
selection of 32 days, she times her commute. The average
commute time for those days was 15.3 minutes. Assume the
population standard deviation is 3.5 minutes and form a 98%
confidence interval to estimate the true mean commute time for
the student.
Here n = 32, , CL = 0.98, So
The margin of error is given by:
The 98% Confidence interval is:
So we are 98% confident that the true mean is between 13.86
minutes and 16.74 minutes.
9. A Confidence Interval
A logistics company claims the true average price for a gallon
of regular, unleaded gas is $3.35. A researcher has recently used
sample data to form a 98% confidence interval estimate of the
true average price of regular, unleaded gas. The interval is
given by $3.26 ± $0.06. Do the results contradict the logistics
company’s claim?
The 98% confidence interval in this case is: ($3.26 - $0.06,
$3.26 + $0.06) = ($3.20, $3.32) which shows that we are 98%
confident that the true average price per gallon of unleaded gas
will be between $3.20 and $3.32.
So $3.35 is outside of this interval, the results of the researcher
does contradict the logistics company’s claim.
On the other hand, if the company had claimed that the average
price per gallon of unleaded gas was $3.31, this would have
been included in the interval and the results would not have
contradicted the company’s claim.
The Factors Affecting the Margin of Error or Width of
Confidence Intervals
The Margin of Error determines how wide our confidence
interval will be.
We do not want wide confidence intervals because narrow
intervals give us a better idea of where the mean lies on the
number line.
There are two ways to reduce the error in a confidence interval:
(1) decrease the confidence level, or (2) increase the sample
size.
Increasing the sample size is not always possible because of
10. costs and implementation considerations.
The population standard deviation is given with the data and
cannot be changed.
7.2
Sample Size Requirements for Estimating the Population Mean
The Formula for Determining the Sample Size Needed to
Estimate the Population Mean
We can use the formula for the Margin of Error to derive a
formula that will tell us the sample size needed to produce a
confidence level that has the particular margin of error and a
desired confidence level. That formula is:
The Special Rounding Rule for Sample Size Calculations
Remember than n represents the number of subjects that were
measured or surveyed in our study, so it cannot be a decimal or
a fraction.
When a decimal occurs, we always want to round up because
rounding up will produce less error in our confidence interval
while rounding down would produce more error.
11. Determining the Sample Size Needed to Estimate the Population
Mean
A stockbroker on Wall Street wants to estimate the average
daily-high price for a stock. What sample size is necessary to
form a 99% confidence interval to estimate the mean daily-high
within 0.50 dollars? Assume the population standard deviation
is known to be 4.59 dollars.
Here CL = 0.99,
So the sample size n is:
, by rounding up
Thus the sample size necessary to form a 99% confidence
interval is 560.
7.3
Interval Estimate of the Population Mean with an Unknown
Population Standard Deviation
Estimating the Mean when the Population Standard Deviation is
Unknown
Often we do not know the population standard deviation (σ)
when attempting to estimate the population mean using a
confidence interval.
When the population standard deviation is unknown, we must
use the sample standard deviation as a substitute
However, since the sample standard deviation is not the same as
the true population standard deviation, we cannot u se the z
distribution to construct the confidence interval.
When the population standard deviation is unknown, we use the
12. t distribution to form our interval estimate of the mean.
Like the z distribution, the t distribution is a bell-shaped
distribution, so we will still need to assume the sample mean
has a normal (or approximately normal) distribution to use the t
distribution.
Estimating the Mean when the Population Standard Deviation is
Unknown
When do we use the t distribution to estimate the population
mean?
When the population standard deviation (σ) is unknown, and we
can assume the distribution of the sample mean is
approximately normally distributed.
The Similarities Between the t and z Distribution
It should be stated at the outset that there is not just one t
distribution, but a family of t distributions.
For every different sample size n, (degree of freedom), there is
a slightly different corresponding t distribution.
These infinitely many t distributions will be defined by their
specific degrees of freedom (n – 1).
The family of t distributions is similar to the standard normal
(z) distribution in several important ways
The most basic similarity between the t distributions and the
standard normal distribution is the fact that they are continuous
distributions.
The shapes of the curves are also similar – both distributions
are symmetric and mound-shaped (i.e., bell-shaped).
The family of t distribution curves and the standard normal
13. curve have the same mean – that mean is zero.
The Similarities Between the t and z Distributions
The diagram contains the graph of the standard normal
distribution and the t distribution for a sample size of 12
The Differences that Exist Between the t and z Curves
The family of t distributions and the standard normal
distribution (z) are similar in three ways: (1) both a continuous
distributions, (2) both are bell-shaped distributions, and (3)
both have a mean of zero.
The differences that exist between the curves all stem from the
fact that they have different standard deviations.
The standard deviation for the standard normal curve is 1,
whereas for the family of t distributions, the standard deviation
varies, but it is always greater than 1.
For every different sample size n (and degree of freedom n – 1),
there is a slightly different standard deviation for the
corresponding t distribution.
This is the only thing that differentiates the otherwise identical
t curves from each other.
14. The Differences that Exist Between the t and z Curves
Since all probability distributions must have a total area of one,
the different standard deviations affect the overall shape of the
curves in a predictive way.
Curves with greater variation (a higher standard deviation) will
be flatter on top and more spread out.
This means that there is more area in the tails and less in the
center of the distribution.
When the curve has less variation (a smaller standard
deviation), it will have more data in the center and small tail
areas.
The Differences that Exist Between the t and z Curves
The closer the standard deviation is to 1, the more the t
distribution will look like the z distribution.
For the family of t distributions, there is an inverse relationship
between sample size (degrees of freedom) and standard
deviation.
As the sample size increases, the corresponding t distributions
have smaller and smaller standard deviations.
This implies that as n increases, the t distributions become more
and more like the standard normal distribution.
The Differences that Exist Between the t and z Curves
15. In the diagram, the two t distributions are graphed along with
the standard normal curve. In comparison to the standard normal
curve, you can see that the two t curves are thicker (i.e., have
more density) in the tails and have less area at the center. The
smaller the sample size the more pronounced these differences
are.
Confidence Interval for Estimating a Population Mean (with
Sigma Unknown)
The formula for the confidence interval to estimate the mean
consists of two values; a lower limit and an upper limit:
Confidence Interval for
where n = the sample size
σ = the population standard deviation
= the sample mean
= the t score separating an area of in the upper tail of the t
distribution with degrees of freedom
Lower LimitUpper Limit
16. Confidence Interval for Estimating a Population Mean (with
Sigma Unknown)
The formula is often expressed as:
However, the most common way to express the formula for the
confidence interval to estimate the mean is
, where
is called the margin of error.
Find Critical t Values
Assuming the population is approximately normal and sigma is
unknown, find the appropriate critical value for a 90%
confidence interval with a sample size of 20.
Since the population is approximately normal, sample size is
small and sigma is unknown, we need to use the t distribution
and hence calculate the t critical value. So here:
CL = 0.90
n = 20
df = n – 1 = 19
So the critical value is: (from the t-table on the next slide)
Find Critical t Values: t Table
17. Form the Margin of Error when Sigma is Unknown
Find the margin of error for a 98% confidence interval estimate
of the population mean when sigma is unknown. The sample
size is 15. The standard deviation is 20.1, and the data appear to
be normally distributed.
Here:
CL = 0.98; n = 15; df = n – 1 = 14; s = 20.1.
So the critical value is: .
So the margin of error is:
The Steps to Create a Confidence Interval (Sigma is Unknown)
The following are the steps to create a confidence interval when
sigma is unknown:
Gather the sample data for the problem, which will include and
the confidence interval.
Find the critical value
Calculate the margin of error (E).
Form the interval by subtracting the margin of error from the
sample mean and adding the margin of error to the sample mean
18. Construct a Confidence Interval when Sigma is Unknown
A waiter wants to know the average amount of time it takes a
table of guests in his section of the restaurant to “turn” (sit,
order, eat, pay, and leave). He times a random selection of 25
tables over several busy nights. For those tables, the average
time to turn was 42.1 minutes. The sample standard deviation
was 4.7 minutes. Assume the turn times are normally
distributed, and form a 90% confidence interval for the true
mean time to turn a table in this waiter’s section of the
restaurant.
Since the population standard deviation is not given and the
distribution of turn times is given to be normal, we use the t
distribution.
CL = 0.90; n = 25; df = n – 1 = 24; s = 4.7. So
, and the margin of error is .
So the 90% confidence interval for the true mean time to turn is
given by
7.4
Interval Estimate of the Population Proportion
Population Proportion
The term proportion refers to the fraction, ratio or percent of
the population having a particular trait of interest.
The symbols for population proportion and sample proportion
are ρ (rho) and (p-hat) respectively.
19. Examples of Population Proportions:
In the United States of America, 16.7% of all babies born have
blue eyes
In 2013, 31.7% of the U.S. population, aged 25 or older, held a
bachelor’s degree or higher.
85% of 18 to 24 year olds, who were raised by at least one
parent having a bachelor’s degree or higher, will attend college.
The Sample Proportion
To calculate the proportion of a sample that has some trait of
interest, we divide the number of subjects (or items) that have
the trait by the number of subjects (or items) belonging to the
sample.
Formula for the sample proportion ():
where
x = the number of subjects (or items) having the trait of interest
n = total number of subjects (or items) sampled
The Sample Proportion
For example, consider the survey results below:
The proportion of students reporting that they earned an A in
Business Statistics is given by:
Number of Survey ParticipantsNumber who earned an A in
20. Business Statistics21520
The Sampling Distribution of the Sample Proportion
Recall that if we randomly select n subjects and x of them have
some trait we are interested in, the sample proportion formed
from the data is:
where x = the number of subjects having the trait we are
interested in.
We use as a point estimate of the population proportion (ρ).
For different samples of size n, a different number (x) of
subjects will have the trait of interest.
This means the value of will vary from sample to sample.
If we want to use it to form an interval estimate of the true
population proportion (ρ), it is important that we know the
sampling distribution of .
The Sampling Distribution of the Sample Proportion
The sampling distribution of
is approximately normally distributed
The expected value (mean) for is the population proportion (ρ).
The standard error for is
can be assumed to be approximately normally distributed when
both and .
We can approximate the standard error of as
21. 45
The Sample Size Requirement for Estimating the Population
Proportion
When constructing a confidence interval to estimate the
population proportion, we can assume is approximately
normally distributed if both and .
Example: A large corporation wants to estimate the proportion
of its part-time employees that would enroll in the company
health insurance plan, if it were made available to them. A
survey of 500 randomly selected part-time employees reveals
that 285 of them would enroll in the plan.
In this example, the sample size is 500 and the number of
employees interested in enrolling in the plan is 285. Using these
quantities, we can calculate the sample proportion ().
The Sample Size Requirement for Estimating the Population
Proportion
Using the sample proportion as an estimate for the population
proportion (ρ), we can check the sample size requirement to
ensure the sampling distribution of the sample proportion is
approximately normal.
and
Since both of the results above are at least 5, it is appropriate to
assume the sampling distribution of the sample proportion is
approximately normally distributed.
22. Formula to Calculate a Confidence Interval for the Population
Proportion
The formula to calculate the confidence interval for the
population proportion ( is given by
where
is the sample proportion
n is the sample size
is the critical value linked to the confidence interval
Constructing a Confidence Interval for the Population
Proportion
An efficiency consultant studied a random selection of 200 e-
mails received by company employees, to determine how many
were relevant to the recipient. Only 36 of the emails were
relevant to their recipients. Form a 95% confidence interval to
estimate the true proportion of relevant emails received by a
typical employee.
Here n = 200, (as only 36 out of the 200 e-mails were relevant),
, CL=0.95, .
So (from the z-table on the following slide) and the margin of
error is:
So the 95% confidence interval estimate is
We are thus 95% confident that the true proportion of relevant
emailed received by a typical employee lies between 0.127 and
0.233.
23. Question: Can we say that it seems that less than a quarter of
emails are relevant? Yes because the upper limit 0.233 < 0.25.
Constructing a Confidence Interval for the Population
Proportion
Interpreting a Confidence Interval for the Population Proportion
The CEO of a logistics company claims that only 5% of its
holiday deliveries arrive late. A 98% confidence interval to
estimate the proportion of late deliveries produced the
following interval: 0.06 to 0.11. Does the interval contradict the
CEO’s claim?
According to the confidence interval estimate, the true
proportion of late deliveries lies between 6% and 11% with 98%
confident.
Since both these numbers are higher than the stated value of 5%
(that is, the interval does not contain 5%), the CEO’s claim is
contradicted.
7.5
Sample Size Requirements for Estimating the Population
24. Proportion
The Formula for Estimating Sample Size for the Population
Proportion
The sample size formula when estimating a population is used
to specify the sample size required to guarantee that your
confidence interval has a certain margin of error and a certain
confidence level.
It is derived by taking the margin of error (E) from the
confidence interval formula for estimating the population
proportion and solving for n.
The equation is:
Since is unknown, we substitute the value 0.5 in the equation
because 0.5(1-0.5) is the maximum, so the value of n obtained
with this value of will be guaranteed to be as large as it
possibly need to be to cover all possible scenarios.
Calculate the Sample Size Needed to Estimate the Population
Proportion
A sales manager at a local car dealership wants to estimate the
proportion of used car sales that include an extended warranty.
What size sample would be needed to estimate the proportion of
extended warranties sold with error of no more than 0.05 and a
confidence level of 99%?
Here E = 0.05, CL=0.99, , and we use
We use the formula:
25. So n = 664, as we always “round up” in case of sample size
determination.
Module 6
point estimators and sampling distributions
Master for Business Statistics
Dane McGuckian
Topics
6.1 Point Estimators and Sampling Distributions
6.2 The Central Limit Theorem
6.1
Point Estimators and Sampling Distributions
Sampling Distributions
The sampling distribution of a statistic is a probability
distribution for all of the possible values of a sample statistic
26. that can be derived from samples of a given size.
Recall that a probability distribution provides all possible
outcomes for an experiment and the probability associated with
each of these outcomes.
Example: If we took every possible random sample of 25 values
from a population and calculated the sample mean for each
sample, the resulting sampling distribution for the sample mean
would provide all possible means that could result from a
sample of 25 values drawn from this population along with the
probability that each of those means occurs.
Depending on the type of data involved, the sampling
distribution can be represented in a table format, as a histogram,
or as a formula.
Sampling Distributions
There are essentially three things we want to know about the
sampling distribution for any sample statistic:
What is the shape of the sampling distribution?
Where is the center (the mean) of the sampling distribution?
How much spread or dispersion (variation) does the sampling
distribution have?
Sampling Distributions
Example: Imagine that we select 2 balls, with replacement, from
a box containing two numbered balls and average the values that
appear on the selected balls. One of the balls has the number 0
printed on it, and the other has the number 1 printed on it. In
this scenario, what is the sampling distribution for the sample
mean?
27. Let’s begin by listing all of the possible outcomes for the two
selections. The possible outcomes are: 00, 01, 10, and 11. Next,
we can determine each of the possible means:
Sampling Distributions
Because each ball has an equal probability of being chosen,
each of the listed outcomes on the previous slide (00, 01, 10,
and 11) has an equal chance of occurring (
Consider the table below:
Next, we will convert this table into a probability distribution
for t he sample mean.
SampleP()0,000.250,10.50.251,00.50.251,110.25
Sampling DistributionP()00.250.50.510.25
Sampling distribution of the Sample Mean
28. Now that we have the probability distribution for the sample
mean, we can use it to calculate the mean of the sample means
and the standard deviation of the sample means:
Point Estimators
A point estimate is a statistic computed from a sample that is
designed to estimate a population parameter.
The preferred estimate for the population mean () is the sample
mean ().
So if we want to estimate the population mean, we would get
some sample data and then we would determine the sample
mean for the sample data, and that would be our point estimate.
The Standard Error of an Estimator
The Standard Error of an Estimator tells us how the estimator
will vary from sample to sample.
The estimator will not be the same for every sample, so the
standard error helps us understand how consistent the estimator
will be from sample to sample.
Population mean:
Point Estimator:
Standard Error:
29. The Desired Traits of a Point Estimator
Ideally, our point estimators should be unbiased estimators.
Among unbiased estimators, we want the estimator with the
minimum variance.
If an unbiased estimator is available they are preferred over
biased estimators.
Example: Estimator A is not unbiased
because it misses almost always.
Estimators B and C are unbiased.
Estimator C has smaller variance than
Estimator B, hence it is called Minimum
Variance Unbiased Estimator (MVUE).
6.2 The Central Limit Theorem
The Central Limit Theorem
The Central Limit Theorem states that for a sufficiently large
sample of size n, taken from a population that is not normally
distributed, the sample mean has an approximately normal
probability distribution.
In most cases, a sample size greater than thirty is large enough
to assume that is approximately normal.
30. The Central Limit Theorem
The Central Limit Theorem describes the sampling distribution
of the sample mean.
If all samples of size n are selected from a population of
measurements with mean, , and standard deviation, , the
distribution of the sample mean has the following mean and
standard deviation (standard error):
Mean of the sample means is:
Standard error of the sample mean (the standard deviation of the
distribution of sample means) is:
The Central Limit Theorem
If the population of measurements is normally distributed, the
distribution of the sample means will be normal regardless of
the size (n) of the sample.
However, if the population of measurements is not normally
distributed, the distribution of the sample means will only be
approximately normal when the sample size is suitably large.
As a good rule of thumb, we will assume that any sample size
larger than 30 is large enough to ensure the distribution for the
sample means is approximately normal.
This approximation will improve for larger values on n.
The Central Limit Theorem
Examples:
The random variable X has a highly skewed distribution. If
samples of size 5 are taken from the population of X values, the
31. distribution of the sample means will not necessarily be normal;
however, if samples of size 35 are taken from the population,
we can assume the distribution of the sample means will be
approximately normal.
The random variable X has a normal distribution. If samples of
size 2 are taken from the population of X values, the
distribution of the sample means will be normal because the
distribution of the sample means is normal at any sample size
when X is normal.
The Mean of the Sample Mean
When discussing the Central Limit Theorem, which describes
the sampling distribution of the sample mean, we stated that the
mean of the sample means for all samples of size n is always
equal to the population mean.
In other words, if all samples of size n are selected from a
population of measurements with mean , the mean of the sample
means is .
Example: If the true average IQ score for a population is 100
and every possible sample of size 15 is taken from the
population, the sample means calculated from each of those
samples will have an average equal to100 because that is the
mean for the population.
For any particular sample size, the mean of all of the sample
means is equal to the population mean.
The Mean of the Sample Mean
The Sample Mean IQ Scores for all Possible Groups of 15
32. People: (this is a partial list because the actual list would be
very long)
To understand the idea discussed on the previous slide, imagine
that for each sample of 15 individuals selected we calculate an
average IQ score.
These sample means will be recorded (perhaps in a list like the
ones illustrated above), and once we have calculated a sample
mean from every possible sample of 15 people, we will then
average all of those sample means in our list.
The result will be the population mean IQ score, which in this
case is 100.
105989711195919410011310195…
The Standard Error of the Mean
When we introduced the Central Limit Theorem, which
describes the sampling distribution of the sample mean, we
discussed the standard error of the mean (the standard deviation
of the sample means) for all samples of size n.
In that discussion we stated that if all samples of size n are
selected from a population of measurements with standard
deviation , the standard error of the mean is .
Example: If the true standard deviation for IQ scores for a
population is 15 and every possible samples of size 9 is taken
from the population, the sample means calculated from each of
those samples will have a standard error that is equal to
For any particular sample size, the standard error for the mean
is equal to the population standard deviation divided by the
square root of the sample size.
33. The Standard Error of the Mean
This definition of the standard error for the mean assumes that
the sampling is done with replacement of that the population we
are sampling from in infinite.
Sampling with replacement implies that a value that has been
selected during the sampling procedure is available to be
selected again and again in the same sample.
In the extreme case, this sampling procedure could produce a
sample of n measurements which consists entirely of one value
repeated n times.
This underlying assumption is a concern, because typically we
do not take samples from infinite populations, and typically, we
do not sample with replacement.
For example, if we are conducting a study on human height by
measuring 10 randomly selected people, we probably would not
want our sample to consist of one person’s height repeated 10
times.
The Standard Error of the Mean
Fortunately, we can modify the standard error formula to
accommodate the finite population case pretty easily.
To find the standard error of the mean when sampling from a
finite population, we use a multiplier often referred to as the
finite population correction factor:
where N is the size of our population and we are selecting a
sample of size n
If we are taking a sample of size n, without replacement,
from a finite population of size N, the standard error for
the mean becomes:
34. The Standard Error of the Mean
The formula for the standard error of the mean when sampling
from a finite population only differs from our previous formula
by the finite population correction factor., and often, we can
ignore this difference.
When sampling from a large finite population without
replacement, it is acceptable to use the original formula we
provided as an approximation to the standard error of the mean.
How large does our finite population have to be to use this
approximation?
Typically, if our sample size is not more than 5 percent of the
population, we can use the population standard deviation
divided by the square root of the sample size to approximate our
standard error for the mean.
For the exercises included in this course, we have assumed that
you will not be using the finite population correction factor.
This means you can safely use the formula provided on the first
slide when you are asked to determine the standard error for the
mean.
The Variation in the Sample Means
If all samples of size n are selected from a population of
measurements with standard deviation , the standard error of the
mean is .
Because the standard error of the mean is equal to the standard
deviation, σ, divided by the square root of the sample size, the
standard error for the mean is always less than the standard
deviation for the random variable.
35. The Variation in the Sample Means
This implies that a set of sample means from a population will
also exhibit less variation than the random variable for that
same population.
For example, if the sample size for the sample means is 4, the
standard deviation for the sample means will be half as large as
the standard deviation () for the random variable.
This means the distribution for the sample means is more
clustered around the mean for the population than the
distribution for the random variable is.
This is a useful trait because it implies that as the sample size
increases, our sample means will move closer and closer to the
true population mean.
The Variation in the Sample Means
Example: An investor has two sets of data involving the closing
stock price for a company in the NASDAQ. One set of data
contains the closing stock prices for a random selection of 12
days taken over the course of the year, and the other set of data
contains 12 averages obtained from random samples of 4 days
of closing prices taken over the same year. Which data exhibits
a larger amount of variation?
36. The Variation in the Sample Means
Closing prices of 12 randomly chosen days (sample standard
deviation s = $85.24):
Average closing prices for 12 samples of n = 4 days (sample
standard deviation s = $38.89):
It is clear that the set of averages has far less dispersion than
the set of individual observations.
If we had every sample mean possible for all samples of four
(selected with replacement), the standard deviation of these
sample means would be , where is the standard deviation for
the daily closing prices.
The Central Limit Theorem and Calculating Probabilities for the
Sample Mean
A software company’s average daily stock price last year was
$38.12. The standard deviation for those prices was $2.45. If a
random selection of 32 days were chosen from last year, what is
the probability that the average price of the company’s stock for
those 32 days is more that $37.00?
By Central Limit Theorem, since the sample size (32) is greater
than 30, the distribution of the average stock prices is
approximately normal. So
The z-score for $37 is:
.
So
37. Module 5
continuous random variables
Master for Business Statistics
Dane McGuckian
Topics
5.1 Continuous Random Variables
5.2 The Normal Distribution
5.3 Applications of the Normal Distribution
5.4 Normal Approximation to the Binomial Distribution
5.1
Continuous Random Variables
Continuous Random Variables
Continuous random variables usually result from measuring
something like a distance, a weight, a length of time, a volume,
or some other similar quantity.
38. Because they can take on any value inside a particular interval,
there are an infinite, uncountable number of possible values for
any continuous random variable.
For this reason, when working with a continuous random
variable, we will discuss the probability that the random
variable is within some specified range.
Continuous Random Variables
Example: A fast-food restaurant manager tracks the length of
time his customers wait or their orders.
The random variable is continuous because it consists of
measured lengths of time.
We could consider the probability that a customer waits more
that five minutes for his or her order, less than five minutes,
between four and five minutes, or some other suitable interval
of time.
But the probability that a customer waits exactly five minutes
for the order is zero.
A consequence of continuous random variables having an
infinite, uncountable set of possible set of values is that the
probability of any continuous random variable equaling a
specific value is always zero.
Continuous Random Variables
The probability distribution for a continuous random variable is
usually represented by a function called the probability density
function (pdf).
These functions produce smooth curves when graphed, and
probability for the random variable is defined as the area under
the curve between any two specified points.
39. Discrete versus Continuous Random Variables
It is common to discuss the probability that a discrete random
variable takes on a specific value, but because a continuous
random variable has an infinite number of possible values in a
particular range, we do not typically discuss the probability that
a continuous random variable takes on a specific value.
Continuous probability distribution
Discrete probability
distribution
The Area under a Continuous Probability Function
In a continuous probability distribution, the probability than an
event, x, is between two numbers is represented by an area, A.
The area under the curve represents all the possible
probabilities that can occur from negative infinity to positive
infinity.
Because the total probability for all continuous probability
distributions is one, the area under the curve must also be one.
40. Continuous Uniform Distribution
A continuous random variable has a uniform distribution if the
graph of its probability distribution is rectangular in shape and
can be completely defined by its minimum and maximum
values.
Like all continuous distributions, the total area under the graph
of a uniform distribution is equal to one, and there is a direct
relationship between the area under the curve between two
specific points and the probability of the random variable
assuming a value between those two points.
The mean for the uniform distribution is
Continuous Uniform Distribution
The standard deviation for the uniform distribution is
where:
is the minimum value for the distribution
is the maximum value for the distribution.
For any uniform distribution, there is a uniform height to its
curve.
The height of the uniform distribution for any value such that
41. is given by .
For any value outside of the interval the height of the curve is
zero.
Continuous Uniform Distribution
Since the shape of the uniform curve is rectangular and
probability corresponds to area under the curve, the probability
that for a uniformly distributed random variable defined by the
interval is
when .
The Probability of a Uniform Distribution
For a uniform distribution defined on the interval
when .
Probabilities for the Uniform Distribution
The amount of time it takes an accountant to prepare tax returns
for her clients is uniformly distributed over the interval between
42. 15 minutes and 60 minutes. What is the probability that she will
finish a tax return in 40 minutes or less?
Here a = c = 15; b = 60, d = 40. So
Thus there is a 55.6% probability that she will finish a tax
return in 40 minutes or less.
5.2 The Normal Distribution
The Normal Distribution
The normal distribution is a continuous distribution that appears
in many applications
Many natural phenomena can be modeled using the normal
probability density function.
The formula for the normal curve is
where
Note: and
The Normal Distribution
43. Notice that the formula for the normal distribution contains the
symbols and , which represent the mean and the standard
deviation respectively.
The mean determines the location of the curve on the number
line, and the standard deviation determines the width or spread
of the distribution.
The values of these parameters depend on the population being
studied.
For this reason, the formula on the previous slide actually
represents a family of normal distributions, not just one curve.
Example: The heights of men have a normal distribution, and
men have mean height of 69 inches.
The heights if women are also normally distributed, but the
mean for women’s heights is 64 inches.
The two curves also have different standard deviations.
The Normal Distribution
In the illustration below, the two normal curves have different
means and different standard deviations.
The difference in the shape of the two curves is a result of the
curves having different standard deviations.
The difference in their position on the number line is due to
their different means.
The taller and the narrower curve belongs to the distribution
with the smaller standard deviation.
The Shape of the Normal Distribution
44. The graph of the normal probability density function is bell-
shaped, but there is not just one normal curve.
There is no limit to the different possible combinations of and ,
so there are an infinite number of different normal curves.
The particular scale and location of a normal distribution will
depend on the distribution’s specific mean and standard
deviation.
However, all normal curves are bell-shaped, and they are always
perfectly symmetric around their mean.
The Shape of the Normal Distribution
The graph of the normal probability distribution function is
bell-shaped and perfectly symmetric around its mean.
This indicates that the left side of the normal distribution is a
perfect mirror image of the right side of the distribution.
Since the total area under all normal curves is 1.00 and all
normal curves are symmetric around their mean, half of the area
(0.50) is below the mean, and half of the area (0.50) is above
the mean.
This is very useful information.
Example: If women’s heights are normally distributed and the
average height for women is 64 inches, we can say with
certainty that half of all women are shorter than 64 inches.
Of course that also implies half of all women are taller than 64
inches.
The Normal Distribution
45. There is not just one normal curve but there are an unlimited
number of normal curves.
Example: Human height is normally distributed, but the heights
of men and women form different normal distributions.
IQ scores are also normally distributed, but those scores form a
different normal curve than the ones formed by male and female
heights.
The list of examples is endless, so when we speak of the normal
distribution, we are referring to a family of curves that have the
same underlying structure.
The mean and standard deviation, and , allow a single
probability density function to produce a family of normal
distributions.
Converting Normal Random Variables into Standard Normal
Random Variables
When working with normal random variables, we have a need to
find areas or probabilities, but the probability density function
for the normal distribution is mathematically difficult to work
with.
For this reason, when solving problems involving a normally
distributed random variable, it would be very helpful to have a
table of probabilities for the normal curve.
However, there isn’t just one normal curve to tabulate
probabilities for. Because there is no limit to the different
possible combinations of and , there are an infinite number of
different normal curves. Therefore, we would need an infinite
number of normal probability tables to handle every possible
application of the normal curve.
Fortunately, there is a way to work around this difficulty.
46. Converting Normal Random Variables into Standard Normal
Random Variables
It is possible to convert any normal random variable with mean
() and standard deviation () into a standard normal random
variable.
A standard normal random variable is a normally distributed
random variable that has a mean equal to zero ( = 0) and a
standard deviation equal to one ( = 1).
To convert a normal random variable () into a standard normal
random variable (), we use the following formula:
where
is the value of a measurement (or observation) taken from a
normally distributed population
is the mean of the distribution for
is the standard deviation of the distribution for
is the standard normal value
Parameters of the Standard Normal Distribution
A standard normal random variable is a normally distributed
random variable that has a mean equal to zero ( = 0) and a
standard deviation equal to one ( = 1).
Because the standard normal distribution has a mean of zero and
47. a standard deviation of one, values equate to the number of
standard deviations above (or below) the mean.
Example: A standard normal value of 1 is the same as one
standard deviation above average.
By using a standard normal probability table, it is possible to
find the probability that a standard normal value falls between
any two points on the –axis.
Z tables and Finding the Areas Under the Standard Normal
Curve Between the Mean and a Value
Use the standard normal curve to find
P(
, because
the normal curve is symmetric.
= 0.4429 from the table.
48. Areas Under the Standard Normal Curve Inside an Interval
Surrounding the Mean
Use the standard normal curve to find
because of symmetry
, from the normal table
= 0.7597
Areas Under the Standard Normal Curve between a Positive Z
Value and Infinity
Use the standard normal curve to find
, as the total area to the left of the mean (0) is 0.5
, from the table
49. Areas Under the Standard Normal Curve between Two Values
on the Same Side of the Mean
Use the standard normal curve to find
Areas Under the Standard Normal Curve between a Negative Z
Value and Infinity
Use the standard normal curve to find
50. , as the total area to the left of the mean (0) is 0.50.
, from the table
5.3
Applications of the Normal Distribution
The Probability that a Non-Standard Normal Random Variable
is Greater than an Above-Average Value
The time it takes a computer chip manufacturer to produce a
single chip is normally distributed with a mean of 18.0 seconds
and a standard deviation of 1.2 seconds. Find the probability
that a chip will take longer than 19.8 seconds to produce.
51. The Probability that a Non-Standard Normal Random Variable
is Less than an Above-Average Value
A large investment bank in Miami released a report on the
starting salary offers it made to MBA graduates. The salaries
are normally distributed with a mean of $89,200 and a standard
deviation of $2,100. Find the probability that a randomly
selected MBA graduate was offered a starting salary of less than
$92,000.
The Probability that a Non-Standard Normal Random Variable
is Between Two Values that Surround the Mean
The containers on a mega-cargo ship in the port of Los Angeles
have weights that are normally distributed with a mean of
55,600 pounds and a standard deviation of 2,800 pounds. What
is the probability that a randomly selected container from the
ship weighs between 53,123 pounds and 60,123 pounds?
52. The Probability that a Non-Standard Normal Random Variable
is Between Two Values that are on the Same Side of the Curve
A manufacturer produces gears for use in an engine’s
transmission that have a mean diameter of 10.00mm and a
standard deviation of 0.03mm. The lengths of these diameters
have a normal distribution. Find the probability that a randomly
selected gear has a diameter between 9.94mm and a 9.96mm.
The Value Corresponding to an Upper Percentile of the Normal
Distribution
A company in California is concerned about the length of time
that its employees spend commuting to work. The one-way
commute times for its employees are normally distributed with a
mean of 32.1 minutes and a standard deviation of 5.3 minutes.
What is the commute time that separates the longest 20% of
commutes from the rest?
53. Here we will work in the “reverse” direction – from % Z
(from table) X (using formula).
The z-value such that
is 0.84 (closest
probability in the table being 0.2995.
Now,
So 36.6 minutes is the commute time that separates the longest
20% of commutes from the rest.
The Value Corresponding to a Lower Percentile of the Normal
Distribution
A financial services company gives an analytical reasoning test
to all job applicants. The completion times for the test are
normally distributed with a mean of 50.40 minutes and a
standard deviation of 3.10 minutes. What completion time
separates the fastest 6% of applicants from the others?
We again work in the “reverse”
direction. Here the z-value such
that
is -1.555 (closest
probability value is 0.4394).
Now
So 45.58 minutes is the completion time that separates the
fastest 6% of applicants from the others.
54. 5.4
Normal Approximation to the Binomial Distribution
Using the Normal Distribution to Approximate the Binomial
Distribution
When using the normal curve to estimate a binomial probability
distribution, we must check two things to confirm the fit is
reasonably good:
If either of these is not true, we need to find a different method
of approximation.
The Use of the Continuity Correction Factor
Continuity correction is used when using the normal
approximation to binomial probability.
Example: The rectangle for x=2 actually goes from 1.5 to 2.5 on
the normal distribution.
Therefore we need to add or subtract that extra 0.5 when we are
looking at the probability that x is less than or greater than 2.
55. Know the Reason for the Use of the Continuity Correction
Factor
A marketing firm for the movie industry reports that the average
film is 128 minutes with a standard deviation of 15 minutes.
Assuming these film durations have a bell-shaped distribution,
what percent of films have a duration between 158 minutes and
173 minutes?
The area marked in red is required, which is given by:
49.85% - 47.5% = 2.35%
Thus, 2.35% of films have a duration between 158 minutes and
173 minutes.
The Use of the Continuity Correction Factor
Based on prior experience, a car dealership has a 45% chance of
selling an extended warranty with each used car that is sold. We
want to use the normal approximation to the binomial
distribution to find the probability of selling 25 or less extended
warranties when 60 cars are sold. Using continuity correction,
state the appropriate probability that will need to be found on
the normal curve.
Here X is the number of warranties to be sold, so X = 25 or less.
n = 60, p = 0.45, and 1-p = q =0.55.
So, on
the bell-shaped curve (normal) is the
probability that will need to be found.
56. The Normal Distribution and the Probability that a Binomial
Random Variable is Greater than a Value
Thirty percent of visitors to a local toy retailer will make a
purchase before exiting the store. Use the normal approximation
for binomial probability to determine the probability that more
than 50 visitors out of 200 will make a purchase.
Here X is the number of visitors who make a purchase, so X =
more than 50. Also n = 200, p = 0.30, and 1-p = q =0.70. It is a
binomial distribution because a customer will either make a
purchase or not.
Mean = = 200.0.30 = 60
Using the continuity correction factor, we have to find because
the problem states “more that 50”. The z-score is:
So, (from table).
The Normal Distribution and the Probability that a Binomial
Random Variable is Less than a Value
A small regional airline overbooks its flights because
historically only 90% of the reservations will actually show up
for the flight. If a flight has 100 available seats, the airline will
typically sell 110 reservations for the flight. What is the
probability that at most 95 people show up for a flight with 110
reservations?
Here X is the number of people who show up so X = at most 95.
Also n = 110, p = 0.90, and 1-p = q =0.10. It is a binomial
distribution because a person will either show up for the flight
57. or not.
Mean = n.p = 110.0.90 = 99
Using the continuity correction factor, we have to find because
the problem states “at most 95”. The z-score is:
So, (from table).
The Normal Distribution and the Probability that a Binomial
Random Variable is Between Two Values
A small regional airline overbooks its flights because
historically only 90% of the reservations will actually show up
for the flight. If a flight has 100 available seats, the airline will
typically sell 110 reservations for the flight. Use the normal
approximation for binomial probability to determine the
probability that between 100 and 107 people (inclusive) show
up for a flight with 110 reservations?
Here X is the number of people who show up so X = between
100 and 107 or [100,107]. Also n = 110, p = 0.90, and 1-p = q
=0.10.
Mean = = 110.0.90 = 99
Using the continuity correction factor, we have to find because
100 and 107 are both included”. The z-scores are:
So, (from table).
Module 4
58. Discrete Random variables
Master for Business Statistics
Dane McGuckian
Topics
4.1 Probability Distributions for Discrete Random Variables
4.2 Expected Value, Variance, and Standard Deviation for
Discrete Random Variables
4.3 The Binomial Probability Distribution
4.4 The Poisson Probability Distribution
4.1
Probability Distributions for Discrete Random Variables
Discrete Random Variable
A discrete random variable is a variable that can only assume a
countable number of values.
The achievable values of a discrete random variable are
separated by gaps.
Example: a publisher may sell 300,000 or 300,001 copies of its
latest book, but it cannot sell 300,000.159 copies of its latest
book
Discrete random variables contain observations that are not
measured on a continuous scale
Most often a discrete random variable contains observations
that are derived from counting something.
59. Discrete Random Variables
Examples of Discrete Random variables:
The number of clicks received by an online advertisement over
the past hour
The number of books sold by an author yesterday
The number of people missing the most recent flight from
Miami to London
The number of parking violations last semester on campus
Discrete Probability Distributions
The probability distribution of a discrete random variables lists
all of the possible outcomes for the random variable and the
associated probability for each of those outcomes.
The distribution can be represented by a table, a graph or a
formula.
Number of Female Jurors, XProbability of Outcome
P(X)00.00810.06120.18630.30340.27850.13660.028
Characteristics of a Probability Distribution
A probability distribution lists all possible outcomes for the
experiment and the corresponding probability for each of those
outcomes.
Remember that the probabilities cannot be negative
Each probability must lie between zero and one
The sum of the probabilities for all of the outcomes must be
one.
60. Example: Probability distribution of the number of free throws
made by basketball players who make free throws 80% of the
time (X). For instance, there is a 4%
chance that a player misses both throws
All these probabilities are non-negative
Each probability lies between 0 and 1
The sum of these probabilities are:
0.04+0.32+0.64 = 1
XP(X)00.0410.3220.64
4.2
Expected Value, Variance, and Standard Deviation for Discrete
Random Variables
The Mean of a Discrete Probability Distribution
The average value for a probability distribution is referred to as
the expected value of the probability distribution.
It represents the long-run typical value for the random variable.
If it were possible to run the trials indefinitely, the expected
value would be the mean for the infinite set of outcomes for the
random variable that would result from those trials.
The Expected value
The expected value is essentially a weighted average of the
possible outcomes for the random variable. The weights are the
corresponding probabilities for those outcomes.
Just as the arithmetic mean we studied earlier, it is common for
the expected value to be a decimal of a fraction even when the
61. original set of outcomes must be whole numbers.
The Expected value of a Discrete Random Variable
How much money on average will an insurance company make
off of a 1-year life insurance policy worth $50,000, if they
charge $1000.00 for the policy and each policy holder has a
0.9999 of surviving the year?
Average implies mean, and that mean “expected value” in the
context of a probability distribution. The formula is:
If a person lives, the company makes
$1000 (hence it is “positive”); if the person
dies, it pays the family $50,000 (but also
gets $1000 from the family), so their loss
is $50,000 - 1000 = $49,000.
EventsXP(X)x.P(x)Lives+10000.9999999.90Dies-49,0001 –
0.9999 = 0.0001-4.901.0000995.00 =
The Expected value of a Discrete Random Variable
A life insurance policy that sells for $200 and should the person
pass away before the end of the year, the family gets a check for
$10,000 from the company.
The company can expect their profits divided by the number of
62. policies sold (profit per policy) to be approximately $190.
The expected value is the long-run average after many, many
trials, so while the company’s average profit is unlikely to ever
be exactly $190, the more policies they sell, the closer and
closer the company’s profit per policy will be to $190.
EventsXP(X)x.P(x)Dies-9,8000.001-
9.80Lives2000.999+199.801.0000$190.00 =
Using Expected value to Distinguish between Two Possible
Courses of Action
A bank can either risk $20,000 on a currency investment that
has a 51% chance of earning them $40,000 in profit, or then can
risk $700,000 on a bond investment that has a 98% chance of
earning them $40,000. In the long run, which strategy will yield
the most profit?
Currency: Bond:
Average profit from currency investment is $10,600 and that
from the bond investment is $25,200. Thus the bond yields
higher average profit, and is the better
choice.xP(x)x.P(x)Profit+40,0000.5120,400Loss-20,0000.49-
9,8001.00$10,600 = xP(x)x.P(x)Profit+40,0000.9839,200Loss-
700,0000.02-14,0001.00$25,2000 =
The Variance and Standard Deviation of a Discrete Random
63. Variable
The mean value for a discrete probability distribution provides
the typical value for the random variable.
In other words, the mean tells us what we can expect to happen
on average over the long run, but if we want to know how
varied the outcomes for the random variable will be, we can
calculate the variance or standard deviation for the random
variable.
Variance for a probability distribution
The standard deviation for the random variable is found by
taking t he square root of the variance of the random variable.
Standard Deviation for a probability distribution
The Variance and Standard Deviation of a Discrete Random
Variable
Calculate the standard deviation of the probability distribution
shown here (round to the thousandths place):
xP(x)x.P(x)000.1500110.480.480.48420.250.501.00930.120.361
.081.34 = 2.56 =
64. Determine if an Event is Unusual using the Mean and Standard
Deviation of a Random Variable
A business venture offers an expected profit of $28,000 with a
standard deviation of $5,250. Would it be unusual to earn less
than $20,000 on the deal? (Hint: consider any value more than
two standard deviations away from the mean as unusual)
Thus an earning of $20,000 is not unusual because it falls
within the interval above.
4.3
The Binomial Probability Distribution
The Five Characteristics of a Binomial Experiment
A Binomial Experiment has a fixed number of trials, only two
possible outcomes for each trial, one trial cannot affect outcome
of the next trial, the probability has to remain constant from one
trial to the next, and x must represent the number of successes.
Example: Flip a coin 3 times and count the number of heads that
turn up (say, 1). Is this a binomial experiment?
There is a fixed number of trials, n = 3
There are 2 outcomes in each trial – heads and tails
The trials are independent, and the outcome of each flip does
65. not affect that of the later flips
Each flip had a 50% chance of turning up heads
x = 1 here (success class is “heads”)
Binomial Probability Formula
The probability of having X successes out of n trials during a
binomial experiment is given by the following formula (recall
that :
where
n = the number of trials for the binomial experiment
x = the number o successes
p = the probability of a success
q = the probability of a failure (
Binomial Probability Formula
Example: If a binomial experiment involves slipping a fair coin
7 times and counting the number of heads that result, the
probability of 5 heads turning up in 7 flips is provided below:
n = 7 (there are 7 flips of the fair coin)
x = 5 (we are looking for the probability of getting 5 heads)
p = 0.50 (a fair coin has a 50% chance of turning up heads on a
single flip)
q = 0.50 (the probability of failure is found by subtracting the
probability of success from 1)
Thus the probability of getting 5 heads out of 7 flips is 1.64%.
66. The Probability of X successes in a Binomial Experiment
A cable company believes that their new promotion will
convince 20% of satellite television subscribers to sign up for
cable. If the company is correct, what is the probability that 2
out of 8 randomly selected satellite users end up switching to
cable after hearing the promotion?
The fact that there are 2 groups that will behave differently
(some will switch and some will not) denotes it’s the binomial
distribution.
Test to determine if this is a binomial experiment: (1) fixed
number of trials = 8; (2) there are 2 possible outcomes (switches
or not); (3) constant probability of success (switching) is 20%;
(4) 8 unique users, so trials are independent (assume no user is
called twice).
n = 8, X = 2 switch, p = 0.20, q = 1 – 0.20 = 0.80. So
Thus there is a 29.4% probability that 2 out of the 8 satellite
users switch to cable after hearing the promotion.
The Probability of a Cumulative Set of Events for a Binomial
Experiment
A drug company reports that 65% of balding men would benefit
from using an over the counter hair-loss solution they
manufacture. Assuming the company’s claim is correct and a
67. random sample of 10 men are selected for a clinical trial of the
product, what is the probability that at least 9 of the men
benefit from the solution?
This is a binomial experiment because: (1) there is a fixed
number of trials, 10 men selected; (2) each trial has 2 outcomes
(working or not working); (3) constant probability of 65% of
benefitting from the solution; (4) trials are independent (no
chance of repetition)
Here n = 10, X = 9 or 10, p = 0.65, q = 1 – 0.65 = 0.35. So
P(X = 9 or X = 10) = P(X = 9) + P(X = 10)
= +
= 0.086
So there is an 8.6% chance that at least 9 men would benefit
from the solution.
The Probability of a Cumulative Set of Events for a Binomial
Experiment
A laptop manufacturer knows that 30% of its laptops will fail
within the first two years of use. If seven randomly selected
customers are surveyed, what is the probability that more than 3
of them experienced a laptop failure within the first two years
of use?
This is a binomial experiment because: (1) there is a fixed
number of trials – 7 men selected randomly; (2) there are two
outcomes (a laptop fails or not within the first two years of
use); (3) constant probability of 30% of a laptop failing in two
years’ time; (4) customers are independent.
Here n= 7, X = more than 3 had a failure (4, 5, 6, or 7), p =
0.30, q = 1 – 0.30 = 0.70. So, from the table:
P(X=4) = 0.0972
P(X=5) = 0.0250
P(X=6) = 0.0036
P(X = 7) = 0.0002
68. Adding all these,
P(X more than 3) = 0.1260
Mean for a Binomial Probability Distribution
We could calculate the mean of the binomial probability
distribution by listing all of the possible outcomes for the
experiment, listing all of their corresponding probabilities, and
then applying the formula:
However, that method could be very time consuming.
There is a simpler approach when trying to find the mean of a
binomial probability distribution.
First we identify the number of trials for the experiment (n) and
the probability of success (p). Then we apply the following
formula:
Mean for a Binomial Probability Distribution
Example: If a quality control manager samples computer chips
from a production line with replacement that have a 0.009
probability of being defective, what is the average number of
defective chips that will be found in a sample of 200 chips?
This sampling procedure produces a binomial probability
69. distribution, so we can apply the formula for finding the mean
of a binomial distribution.
In this example, there are 200 trials, and the probability of
finding a defective chip in 0.009
defective chips
For groups of 200 computer chips random selected from the
production line, the average number of defective chips is 1.8.
The Mean of a Binomial Random Variable
An electronics retailer notes that only 8% of its online
customers choose to purchase their extended service plan. If the
retailer has 300 online sales over the next month, what is the
expected number of customers that will choose to purchase the
extended service plan?
This satisfies the conditions of a binomial experiment. So
since n = 300 and p = 0.08 here.
Thus the expected number of customers that will choose to
purchase the extended service plan is 24.
Standard Deviation for a Binomial Probability Distribution
We could calculate the standard deviation for the binomial
probability distribution by listing all of their corresponding
probabilities, and then applying the formula
.
However, that method would be very time consuming.
70. There is a simpler approach when trying to find the standard
deviation of a binomial probability distribution.
First, we identify the number of trials for the experiment (n)
and the probability of success (p).
Then we apply the following formula:
Standard Deviation for a Binomial Probability Distribution
Example: If a quality control manager samples computer chips
from a production line with replacement that have a 0.009
probability of being defective, what is the standard deviation
for the number of defective chips that will be found in a sample
of 200 chips?
This sampling procedure produces a binomial probability
distribution, so we can apply the formula for finding the
standard deviation of a binomial distribution.
In this example, there are 200 trials, and the probability of
finding a defective chip in 0.009.
n = 200, p = 0.009, q = 1 – 0.009 = 0.991
Standard Deviation for a Binomial Probability Distribution
The variance will be
Then we simply take the square root to find the standard
deviation:
71. defective chips
For groups of 200 computer chips randomly selected from this
factory, the standard deviation for the number of defective chips
is 1.336.
Standard Deviation of a Binomial Random Variable
A recent report states that only 28% of software projects were
expected to finish on time and on budget. If we randomly
sample 80 software projects, what is the standard deviation for
the number of projects that are expected to finish on time and
on budget?
This satisfies all the conditions for binomial experiment, so we
can use the formula for the standard deviation to calculate this:
4.4
The Poisson Probability Distribution
The Poisson Distribution
The Poisson distribution is a discrete probability distribution
that provides probabilities for the number of occurrences of
some event over a given period, interval, distance, or space.
Example: A customer service call center might use the Poisson
72. distribution to describe the behavior of incoming calls over
different time periods.
Example: A website might use the Poisson distribution to
estimate the likelihood of some number of individuals logging
onto the site between the hours of 12:00AM and 1:00AM.
Example: A mining company might use the Poisson distribution
to model the number of methane gas releases over a specified
depth.
The Poisson distribution is typically used to model the
occurrences of rare events.
The Poisson Distribution
The probability that the specified events occurs X times over
some defined interval is given by the following formula:
where
(mu) is the mean (expected) number of occurrences (successes)
over a particular interval
x is the number of occurrences (successes)
e is a constant (the base of the natural log) that is approximately
equal to 2.71828
The Poisson Distribution
Here are some important characteristics of the Poisson
distribution:
The random variable is the number of occurrences of some
event over some defined interval.
The probability of the event is proportional to the size of the
closed interval.
The intervals do not overlap.
73. The occurrences are independent of each other.
Mean and Standard Deviation of the Poisson Distribution
The mean of the Poisson distribution is
The standard deviation for the Poisson distribution is given by
Some Important Differences between the Binomial and the
Poisson Distributions
The binomial distribution is dependent upon the sample size and
the probability of success, while the Poisson distribution only
depends on the mean .
In the binomial distribution, the random variable can take on
values of 0,1,..,n, while in the Poisson distribution, the random
variable can be any integer greater than or equal to zero.
In other words, there is no upper bound for the number of
occurrences in the given interval.
Probability and the Poisson Distribution
A cellular communication company finds that during a 10-
minute phone call there will typically be one incidence of poor
reception. Use a Poisson distribution to calculate the probability
that there will be 5 incidences of poor reception during a call
that lasts an hour.
74. Here x = 5, = 6 since an average of 1 incidence of poor
reception in a 10-minute period implies an average of 6
incidences in 60 mins (1 hour).