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Chapter 4: Probability
4.3: Complements and Conditional Probability, and Bayes' Theorem
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Chapter 5: Discrete Probability Distribution
5.2 - Binomial Probability Distributions
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Chapter 5: Discrete Probability Distribution
5.1: Probability Distribution
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Chapter 2: Exploring Data with Tables and Graphs
2.3: Graphs that Enlighten and Graphs that Deceive
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Chapter 4: Probability
4.3: Complements and Conditional Probability, and Bayes' Theorem
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Chapter 5: Discrete Probability Distribution
5.2 - Binomial Probability Distributions
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Chapter 5: Discrete Probability Distribution
5.1: Probability Distribution
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Chapter 2: Exploring Data with Tables and Graphs
2.3: Graphs that Enlighten and Graphs that Deceive
Similar to Quantitative Methods for Lawyers - Class #9 - Bayes Theorem (Part 2), Skewness, Kurtosis & Data Distributions - Professor Daniel Martin Katz
My name is Moses Alex. I am associated with statisticshomeworkhelper.com for the past 15 years and have been assisting the statistics students with their homework.
I have a Masters in Statistics from Leeds Trinity University.
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 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.
Standard Error & Confidence Intervals.pptxhanyiasimple
Certainly! Let's delve into the concept of **standard error**.
## What Is Standard Error?
The **standard error (SE)** is a statistical measure that quantifies the **variability** between a sample statistic (such as the mean) and the corresponding population parameter. Specifically, it estimates how much the sample mean would **vary** if we were to repeat the study using **new samples** from the same population. Here are the key points:
1. **Purpose**: Standard error helps us understand how well our **sample data** represents the entire population. Even with **probability sampling**, where elements are randomly selected, some **sampling error** remains. Calculating the standard error allows us to estimate the representativeness of our sample and draw valid conclusions.
2. **High vs. Low Standard Error**:
- **High Standard Error**: Indicates that sample means are **widely spread** around the population mean. In other words, the sample may not closely represent the population.
- **Low Standard Error**: Suggests that sample means are **closely distributed** around the population mean, indicating that the sample is representative of the population.
3. **Decreasing Standard Error**:
- To decrease the standard error, **increase the sample size**. Using a large, random sample minimizes **sampling bias** and provides a more accurate estimate of the population parameter.
## Standard Error vs. Standard Deviation
- **Standard Deviation (SD)**: Describes variability **within a single sample**. It can be calculated directly from sample data.
- **Standard Error (SE)**: Estimates variability across **multiple samples** from the same population. It is an **inferential statistic** that can only be estimated (unless the true population parameter is known).
### Example:
Suppose we have a random sample of 200 students, and we calculate the mean math SAT score to be 550. In this case:
- **Sample**: The 200 students
- **Population**: All test takers in the region
The standard error helps us understand how well this sample represents the entire population's math SAT scores.
Remember, the standard error is crucial for making valid statistical inferences. By understanding it, researchers can confidently draw conclusions based on sample data. 📊🔍
If you need further clarification or have additional questions, feel free to ask! 😊
---
I've provided a concise explanation of standard error, emphasizing its importance in statistical analysis. If you'd like more details or specific examples, feel free to ask! ¹²³⁴
Source: Conversation with Copilot, 5/31/2024
(1) What Is Standard Error? | How to Calculate (Guide with Examples) - Scribbr. https://www.scribbr.com/statistics/standard-error/.
(2) Standard Error (SE) Definition: Standard Deviation in ... - Investopedia. https://www.investopedia.com/terms/s/standard-error.asp.
(3) Standard error Definition & Meaning - Merriam-Webster. https://www.merriam-webster.com/dictionary/standard%20error.
(4) Standard err
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
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 ...
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LexPredict - Empowering the Future of Legal Decision MakingDaniel Katz
LexPredict is an enterprise legal technology and consulting firm, specializing in the application of best-in-class processes and technologies from the technology, financial services, and logistics industries to the practice of law, compliance, insurance, and risk management.
We focus on the goals of prediction, optimization, and risk management to enable holistic organizational changes that empower legal decision-making.
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The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
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Quantitative Methods for Lawyers - Class #9 - Bayes Theorem (Part 2), Skewness, Kurtosis & Data Distributions - Professor Daniel Martin Katz
1. Quantitative
Methods
for
Lawyers
Bayes Theorem (Part 2),
Skewness, Kurtosis
& Data Distributions
Class #9
@ computational
computationallegalstudies.com
professor daniel martin katz danielmartinkatz.com
lexpredict.com slideshare.net/DanielKatz
2. Example:
Marie is getting married tomorrow, at an outdoor ceremony in the
desert.
In recent years, it has rained only 5 days each year.
Unfortunately, the weatherman has predicted rain for tomorrow.
When it actually rains, the weatherman correctly forecasts rain 90% of
the time.
When it doesn't rain, he incorrectly forecasts rain 10% of the time.
What is the probability that it will rain on the day of Marie's wedding?
Bayes Rule
3. Solution: The sample space is defined by two mutually-exclusive
events - it rains or it does not rain.
Additionally, a third event occurs when the weatherman predicts
rain. Notation for these events appears below.
• Event A1. It rains on Marie's wedding.
• Event A2. It does not rain on Marie's wedding
• Event B. The weatherman predicts rain.
Bayes Rule
4. • Event A1. It rains on Marie's wedding.
• Event A2. It does not rain on Marie's wedding
• Event B. The weatherman predicts rain.
In terms of probabilities, we know the following:
• P( A1 ) = 5/365 =0.014 [rains = 5 days per year]
• P( A2 ) = 360/365 = 0.986 [Not rain = 360 days per year]
• P( B | A1 ) = 0.9
[When it rains, the weatherman predicts rain 90% of the time]
• P( B | A2 ) = 0.1 [When it does not rain, the weatherman predicts
rain 10% of the time]
Bayes Rule
6. We want to know P( A1 | B ), the probability it will rain on the day of Marie's
wedding, given a forecast for rain by the weatherman. The answer can be
determined from Bayes' theorem, as shown below:
P( A1 | B ) = _____________P( A1 ) P( B | A1 )_________
P( A1 ) P( B | A1 ) + P( A2 ) P( B | A2 )
P( A1 | B ) =
___________(0.014)(0.9)__________
[ (0.014)(0.9) + (0.1) (0.986) ]
P( A1 | B ) = .1133
Note the somewhat unintuitive result. Even when the weatherman predicts rain, it
only rains only about 11% of the time.
Bayes Rule
7. What Can We Say About The Weatherman?
Bayes Rule
Likelihood Increased from ~1% to ~11%
That is a 11 fold increase in the likelihood
However, it is still pretty unlikely to rain
8. Bayes Rule
How Much Signal / Information ?
We Could Consider a Complex Version of the problem -
Weatherman Predicts Rain + It is the Monsoon Season
Compound Events
The Signal was of limited value because ratio of Type I to
Type II error was not favorable
10. Bayes Rule
Imagine a particular test:
correctly identifies those with a certain disease 94% of the time
and
correctly diagnoses those without the disease 98% of the time
A friend has just informed you that he has received a positive result
and asks for your advice about how to interpret these probabilities.
Before attempting to address your friend’s concern, you research
the illness and discover that 4% of men have this disease.
What is the probability your friend actually has the disease?
11. Define the events:
Express the given information and question in probability notation:
“test correctly identifies those with a certain serious disease 94% of the time”
“test correctly diagnoses those without the disease 98% of the time”
“you discover that 4% of men have this disease”
this statement also tells us that 96% of men do not have the disease
Bayes
Rule
( )1 0.94P B A⇒ = !
1
2
a man has this disease
a man does not have this disease
positive test result
negative test resultC
A
A
B
B
=
=
=
=
!
( )2 0.98C
P B A⇒ = !
( )1 0.04P A⇒ = !
( )2 0.96P A⇒ = !
12. Key Question:
“Given a positive result, What is the
probability your friend actually has
the disease ?”
( )1 ?P A B⇒ = !
13. Bayes Rule
a tree diagram:
!
1
2
a man has this disease
a man does not have this disease
positive test result
negative test resultC
A
A
B
B
=
=
=
=
!
14. Use Bayes’ Theorem and your tree diagram to answer the question:
There is a 66.2% probability that he actually has the disease. The probability is
high, but considerably lower than your friend feared.
Bayes Rule
( )
( ) ( )
( ) ( ) ( ) ( )
1 2
1
1 2 2 2
0.0376
0.662
0.0376 0.0192
P A P B A
P A B
P A P B A P A P B A
⋅
= = ≈
+⋅ + ⋅
!
18. Why Sample?
Might Be Impossible to Get the Full Population
Cost of Getting Full Population
Sampling is concerned with the selection of a subset of
individuals from within a population to estimate
characteristics of the whole population
Sampling
Focus Upon Improving Precision v. Size
19. (1) Defining the population of concern
(2) Specifying a sampling frame, a set of items or events
possible to measure
(3) Specifying a sampling method for selecting items or
events from the frame
(4) Determining the sample size
(5) Implementing the sampling plan
(6) Sampling and data collecting
Sampling Stages
20. Determining the Sample Size
Conceptually We Understand that in order to
obtain a representative sample we need to acquire
somewhere between
1 > ? > Full Population
But Exactly How Many Observations do we need?
21. Random Sampling Error
Imagine a Political Poll
When You Sample at Random It is Possible to
Have a Skewed Set of Observation in the
Sample
where the population of interest are actual voters.
pollsters take smaller samples that are intended to be
representative, that is, a random sample of the population.
It is possible that pollsters sample 1,013 voters who happen to vote
for Bush when in fact the population is evenly split between
Candidate 1 and Candidate 2, but this is extremely unlikely
(p = 2−1013
≈ 1.1 × 10−305
) given that the sample is random.
22. Random Sampling Error
For Simple Random Sample on a large
population, the Inverse of the Square Root of
the Sample Size
23. Random Sampling Error
For Simple Random Sample on a large
population, the Inverse of the Square Root of
the Sample Size
Very
Typically
Reported
24. Random Sampling Error
a random sample of size 400 will give a margin of error, at a
95% confidence level, of 0.98/20 or 0.049 - just under 5%.
For Simple Random Sample on a large
population, the Inverse of the Square Root of
the Sample Size
Example:
25. Random Sampling Error
a random sample of size 400 will give a margin of error, at a
95% confidence level, of 0.98/20 or 0.049 - just under 5%.
a random sample of size 1600 will give a margin of error of
0.98/40, or 0.0245 - just under 2.5%.
For Simple Random Sample on a large
population, the Inverse of the Square Root of
the Sample Size
Example:
26. Random Sampling Error
a random sample of size 400 will give a margin of error, at a
95% confidence level, of 0.98/20 or 0.049 - just under 5%.
A random sample of size 1600 will give a margin of error of
0.98/40, or 0.0245 - just under 2.5%.
For Simple Random Sample on a large
population, the Inverse of the Square Root of
the Sample Size
Example:
Notice: Double the Precision Requires
four times the Sample Size!
27. Top portion of this graphic depicts the relative likelihood
that the "true" percentage is in a particular area given a
reported percentage of 50%.
In other words, for each sample
size, one is 95% confident that the
"true" percentage is in the region
indicated by the corresponding
segment.
The larger the sample is, the
smaller the margin of error.
The bottom portion shows
95% confidence intervals
(horizontal line segments),
the corresponding margins of
error (on the left), and sample
sizes (on the right).
28. Central Limit Theorem
Try this yourself: “Netlogo Central Limit Theorem”
http://ccl.northwestern.edu/netlogo/models/run.cgi?CentralLimitTheorem.715.627
29. Thinking of Data as
a Distribution: Histogram
Histogram - histogram is a graphical representation showing a
visual impression of the distribution of data
(1) consists of tabular frequencies, shown as adjacent
rectangles, erected over discrete intervals (bins)
(2) The height of a rectangle is also equal to the frequency
density of the interval, i.e., the frequency divided by the width
of the interval
(3) Total area of the histogram is equal to the number of data
30. Thinking of Data as
a Distribution: Histogram
Histogram of travel time, US 2000 census. Area under the curve equals
the total number of cases. This diagram uses Q/width from the table.
33. Data as a Distribution
Try to Start Thinking of Any Data Set as a Distribution
This allows you take a broader perspective about the
observations contained therein
When you get a new dataset you should generate some
summary statistics such as
(1) Measures of Central Tendency
(2) Measures of Variation
( including the first four moments of the distribution)
34. Thinking of Data as
a Distribution
Moment 1 = Mean
Moment 2 = Variance
Moment 3 = Skewness
Moment 4 = Kurtosis
38. a negative skew indicates that the tail on the left side
of the probability density function is longer than the
right side and the bulk of the values (possibly
including the median) lie to the right of the mean.
Skewness
39. Skewness
A positive skew indicates that the tail on the right side is
longer than the left side and the bulk of the values lie to
the left of the mean.
40. Calculating Skewness
1. Subtract Mean
from each Raw Score.
Aka, Deviations from
the mean
2. Raise each of these
deviations from the
mean to the third power
and sum. Aka: Sum of
third moment deviations
3. Calculate skewness, which is the sum of the deviations from the
mean, raised to the third power, divided by number of cases
minus 1, times the standard deviation raised to the third power.
41. Calculating Skewness
Try This Problem:
http://www.indiana.edu/~educy520/
sec5982/week_12/skewness_demo.pdf
1. Subtract Mean
from each Raw Score.
Aka, Deviations from
the mean
2. Raise each of these
deviations from the
mean to the third power
and sum. Aka: Sum of
third moment deviations
3. Calculate skewness, which is the sum of the
deviations from the mean, raised to the third
power, divided by number of cases minus 1,
times the standard deviation raised to the third
power.
44. kurtosis is any measure of the "peakedness" of a
distribution
A high kurtosis distribution has a sharper peak
and longer, fatter tails, while a low kurtosis
distribution has a more rounded peak and
shorter, thinner tails.
Kurtosis
45. Distributions with zero excess kurtosis are called mesokurtic, or
mesokurtotic. The most prominent example of a mesokurtic
distribution is the normal distribution
A distribution with positive excess kurtosis is called leptokurtic, or
leptokurtotic. "Lepto-" means "slender". In terms of shape, a
leptokurtic distribution has a more acute peak around the mean and
fatter tails.
A distribution with negative excess kurtosis is called platykurtic, or
platykurtotic. "Platy-" means "broad". In terms of shape, a
platykurtic distribution has a lower, wider peak around the mean and
thinner tails.
Kurtosis
46. The moment coefficient of kurtosis of a data set is
computed almost the same way as the coefficient of
skewness:
and
“excess” kurtosis: = Kurtosis − 3
Calculating Kurtosis
Note: the excess kurtosis is
generally used because the
excess kurtosis of a normal
distribution is 0.