February 17 Blackboards
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The document discusses confidence intervals for means and proportions. It provides formulas for calculating confidence intervals for a mean when sample sizes are both larger and smaller than 30, and the population standard deviation is both known and unknown. It also provides the formula for calculating a confidence interval for a sample proportion when the sample size is large enough. Examples are given of finding confidence intervals for means, proportions, variances, and standard deviations. The margin of error is defined as half the width of the confidence interval.
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This document discusses confidence intervals and how they are used to estimate population parameters from sample data. Some key points:
- Confidence intervals provide a range of values that is likely to include an unknown population parameter, rather than just a single point estimate. They indicate the reliability of an estimate.
- The formula for a confidence interval is point estimate ± (critical value)(standard error). It depends on the sample size, standard deviation, and desired confidence level.
- When the population standard deviation is unknown, the student's t-distribution is used instead of the normal distribution to calculate confidence intervals.
- Sample size calculations can determine the required sample size needed to estimate a population mean within a specified margin of
Violating Np More Than 10
1. The document examines the condition for constructing a one-proportion z-interval that np and n(1-p) should each be greater than or equal to 10. It discusses what happens when this condition is violated through simulations.
2. The simulations showed that when n=20 and p=0.1 (violating the condition), the percentage of intervals capturing the true population proportion p of 0.1 was less than the stated 95% confidence level.
3. There are alternative interval construction methods that better match the claimed confidence level when the original condition is violated, such as using an adjusted sample proportion.
Bca admission in india
This document discusses confidence intervals and how they can be used to estimate population parameters from sample data. It provides the following key points:
- Confidence intervals provide a range of values that is likely to include an unknown population parameter, unlike a point estimate which is a single value. They indicate the reliability of an estimate.
- The formula for a confidence interval of a population mean takes the sample mean and adds/subtracts a critical value times the standard error.
- When the population standard deviation is unknown, the student's t-distribution must be used instead of the normal distribution, as it accounts for the extra uncertainty of estimating the standard deviation from a sample.
- Sample size calculations can determine the
Lesson04_Static11
This document discusses statistical confidence interval estimation. It covers:
1) Confidence interval estimation for the mean when the population standard deviation is known and unknown.
2) Confidence interval estimation for the proportion.
3) Factors that affect confidence interval width like data variation, sample size, and confidence level.
4) How to estimate sample sizes needed to estimate a population mean or proportion within a given level of precision and confidence.
Lesson04_new
This document discusses various methods for constructing confidence intervals to estimate population parameters using sample statistics. It covers confidence interval estimation for the mean when the population standard deviation is known and unknown, estimation for the proportion, and addresses situations involving finite populations. Factors that influence confidence interval width and formulas for determining necessary sample sizes are also presented. Examples are provided to illustrate how to set up confidence intervals and calculate required sample sizes.
L10 confidence intervals
This document discusses confidence intervals for estimating population means from sample data. It begins by explaining how to calculate point estimates and confidence intervals when the sample size is large (n ≥ 30) using the normal distribution. It then covers calculating confidence intervals when the sample size is small (n < 30) using the t-distribution. The key steps covered are determining the appropriate distribution to use based on sample size and knowledge of the population standard deviation, finding the critical values and margin of error, and calculating the confidence interval. Examples are provided to demonstrate how to construct confidence intervals in different situations.
Calculating p value
When you perform a hypothesis test in statistics, a p-value helps you determine the significance of your results. ... The p-value is a number between 0 and 1 and interpreted in the following way: A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, so you reject the null hypothesis.
7. the t distribution
1. Illustrate the t-distribution.
2. Construct the t-distribution.
3. Identify regions under the t-distribution corresponding to different values.
4. Identify percentiles using the t-table.
Visit the website for more services it can offer: https://cristinamontenegro92.wixsite.com/onevs
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Mca admission in india
This document discusses confidence intervals and how they are used to estimate population parameters from sample data. Some key points:
- Confidence intervals provide a range of values that is likely to include an unknown population parameter, rather than just a single point estimate. They indicate the reliability of an estimate.
- The formula for a confidence interval is point estimate ± (critical value)(standard error). It depends on the sample size, standard deviation, and desired confidence level.
- When the population standard deviation is unknown, the student's t-distribution is used instead of the normal distribution to calculate confidence intervals.
- Sample size calculations can determine the required sample size needed to estimate a population mean within a specified margin of
Violating Np More Than 10
1. The document examines the condition for constructing a one-proportion z-interval that np and n(1-p) should each be greater than or equal to 10. It discusses what happens when this condition is violated through simulations.
2. The simulations showed that when n=20 and p=0.1 (violating the condition), the percentage of intervals capturing the true population proportion p of 0.1 was less than the stated 95% confidence level.
3. There are alternative interval construction methods that better match the claimed confidence level when the original condition is violated, such as using an adjusted sample proportion.
Bca admission in india
This document discusses confidence intervals and how they can be used to estimate population parameters from sample data. It provides the following key points:
- Confidence intervals provide a range of values that is likely to include an unknown population parameter, unlike a point estimate which is a single value. They indicate the reliability of an estimate.
- The formula for a confidence interval of a population mean takes the sample mean and adds/subtracts a critical value times the standard error.
- When the population standard deviation is unknown, the student's t-distribution must be used instead of the normal distribution, as it accounts for the extra uncertainty of estimating the standard deviation from a sample.
- Sample size calculations can determine the
Lesson04_Static11
This document discusses statistical confidence interval estimation. It covers:
1) Confidence interval estimation for the mean when the population standard deviation is known and unknown.
2) Confidence interval estimation for the proportion.
3) Factors that affect confidence interval width like data variation, sample size, and confidence level.
4) How to estimate sample sizes needed to estimate a population mean or proportion within a given level of precision and confidence.
Lesson04_new
This document discusses various methods for constructing confidence intervals to estimate population parameters using sample statistics. It covers confidence interval estimation for the mean when the population standard deviation is known and unknown, estimation for the proportion, and addresses situations involving finite populations. Factors that influence confidence interval width and formulas for determining necessary sample sizes are also presented. Examples are provided to illustrate how to set up confidence intervals and calculate required sample sizes.
L10 confidence intervals
This document discusses confidence intervals for estimating population means from sample data. It begins by explaining how to calculate point estimates and confidence intervals when the sample size is large (n ≥ 30) using the normal distribution. It then covers calculating confidence intervals when the sample size is small (n < 30) using the t-distribution. The key steps covered are determining the appropriate distribution to use based on sample size and knowledge of the population standard deviation, finding the critical values and margin of error, and calculating the confidence interval. Examples are provided to demonstrate how to construct confidence intervals in different situations.
Calculating p value
When you perform a hypothesis test in statistics, a p-value helps you determine the significance of your results. ... The p-value is a number between 0 and 1 and interpreted in the following way: A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, so you reject the null hypothesis.
7. the t distribution
1. Illustrate the t-distribution.
2. Construct the t-distribution.
3. Identify regions under the t-distribution corresponding to different values.
4. Identify percentiles using the t-table.
Visit the website for more services it can offer: https://cristinamontenegro92.wixsite.com/onevs
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This document discusses statistical techniques for comparing two sample means, including calculating a confidence interval for the difference between two means to determine if they could be the same or different, and provides examples of how to perform these calculations and interpret the results.
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Confidence Interval ModuleOne of the key concepts of statist.docx
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.
Estimating population values ppt @ bec doms
This document discusses confidence intervals for estimating population parameters. It covers confidence intervals for the mean when the population standard deviation is known and unknown, as well as confidence intervals for the population proportion. Key points include:
- A confidence interval provides a range of plausible values for an unknown population parameter based on a sample statistic.
- The margin of error and confidence level affect the width of a confidence interval.
- The t-distribution is used instead of the normal when the population standard deviation is unknown.
- Sample size formulas allow determining the required sample size to estimate a population parameter within a specified margin of error and confidence level.
Standard Error & Confidence Intervals.pptx
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
Confidence Intervals
This document discusses confidence intervals for population means and proportions. It explains how to construct confidence intervals using the normal distribution for large sample sizes (n ≥ 30) and the t-distribution for small sample sizes. Formulas are provided for calculating margin of error and determining necessary sample size. Guidelines are given for determining whether to use the normal or t-distribution based on sample size and characteristics. Confidence intervals can be constructed for variance and standard deviation using the chi-square distribution.
How to compute for sample size.pptx
To calculate the required sample size for a survey, you need to determine the population size, margin of error, confidence level, and standard deviation. The formula used is sample size = (z-score)2 * p(1-p) / (margin of error)2. For a survey estimating smartphone ownership in a city with a 99% confidence level, 0.03 margin of error, and 0.70 estimated standard deviation, the required sample size is 2,400.
Inferential statistics
The document discusses key concepts in statistical inference including estimation, confidence intervals, hypothesis testing, and types of errors. It provides examples and formulas for estimating population means from sample data, calculating confidence intervals, stating the null and alternative hypotheses, and making decisions to accept or reject the null hypothesis based on a significance level.
lecture13MWF.pdflecture13MWF.pdflecture13MWF.pdf
This document discusses confidence intervals and margin of error in statistical analysis. It defines key terminology like population mean, sample mean, standard deviation, and standard error. It explains that the margin of error depends on the sample size and standard deviation, and provides the formula for calculating sample size needed to achieve a given margin of error. Several examples are provided to illustrate how to construct confidence intervals and determine required sample sizes.
Chapter 6 part1- Introduction to Inference-Estimating with Confidence (Introd...
Introduction to Inference, Estimating with Confidence, Inference, Statistical Confidence, Confidence Intervals, Confidence Interval for a Population Mean, Choosing the Sample Size
Lect 10 Sample Size Estimation.ppt
- The document discusses factors to consider when determining sample size for quantitative studies, including desired level of confidence, margin of error, and how dispersed the population is.
- Sample size is a balance between what is desirable and feasible, and too small a sample may miss important effects or be too imprecise, while too large wastes resources.
- Formulas are provided to calculate the minimum required sample size for estimating a single population proportion based on the expected proportion, desired margin of error, and confidence level. Adjustments may be needed if the population is small.
Section 7 Analyzing our Marketing Test, Survey Results .docx
Section 7
Analyzing our Marketing Test, Survey Results
and Other Metrics Using Confidence Intervals
Rhonda Knehans Drake
Associate Professor, New York University
Data Analytics, Interpretation and Reporting
copyright © 2013
2
• When we estimate population averages or percentages based on
samples, a certain amount of error is present.
• The amount of error present is a function mostly of your
sample size.
• The larger the sample, the less error in your estimates.
• Today we will learn how to place bounds around our estimates
obtained.
• With such an interval, we will then be able to say with 90%, 95%
or 99% confidence that the true population estimate will lie in
these bounds.
Introduction
3
The eight confidence interval formulas we will discuss are for
the following situations:
1. Average or mean based on large samples
2. Average or mean based on small samples
3. Response rate or survey percent based on large samples
4. Response rate or survey percent based on small samples
5. Difference between 2 averages for large samples
6. Difference between 2 averages for small samples
7. Difference between 2 response rates for large samples
8. Difference between 2 response rates for small sample
Confidence Intervals
A-B Split Tests
4
1. Confidence Interval for Averages or Means Based on
Large Samples (n ≥ 30)
To calculate a confidence interval around a mean, the following
information is required:
– The sample mean x obtained from the test.
– The sample standard deviation S obtained from the test.
Many software packages, including Microsoft ExcelTM, can
automatically calculate this value for you. (Review Section 3 for the
standard deviation formula.)
– The sample size n of the test.
This is the number of observations used to calculate your mean. The
sample size must be greater than or equal to 30 in size.
– The desired confidence level: 90%, 95% or 99%.
A confidence interval constructed around the mean will guarantee, with
your specified level of confidence, the true mean will fall within those
bounds.
Sample Means
(Large Sample)
5
• Once all information is known, you construct the confidence
interval around the mean by adding and subtracting from the
mean a multiple of your standard deviation associated with the
sample mean. The “multiple” depends on your desired
confidence level.
• The formula for a confidence interval around the mean is
calculated as follows:
x (Z )(S/n )
Where:
• S is the standard deviation associated with the sample.
• n is the sample size.
• Z is equal to 1.645, 1.96 or 2.575 for a 90%, 95% or 99%
confidence level
Sample Means
(Large Sample)
6
Example:
Money magazine conducts a survey of 100 retirees across the US
and asks them how much they have in their retirement fund.
You obtain an average of $84.75 with a standard deviation of
$18.75, both in thousands of doll.
Statistik 1 7 estimasi & ci
This document outlines key concepts related to estimation and confidence intervals. It defines point estimates as single values used to estimate population parameters and interval estimates as ranges of values within which the population parameter is expected to occur. Confidence intervals provide an interval range based on sample observations within which the population parameter is expected to fall at a specified confidence level, such as 95% or 99%. The document discusses how to construct confidence intervals for the population mean when the population standard deviation is known or unknown.
Sample size
This document explains how to calculate sample size for a survey or poll. It outlines the key factors needed: population size, margin of error, confidence level, and standard deviation. It provides the formula for calculating sample size and works through an example using a 95% confidence level, 0.5 standard deviation, and +/- 5% margin of error. Based on these parameters, the sample size needed is 385 respondents. The document advises adjusting the confidence level or margin of error if the calculated sample size is too large.
Sampling Theory.pdf
The document discusses various sampling techniques for research. It defines key concepts like population, sample, sampling frame and sampling methods. It explains that the sampling process involves defining the target population, choosing a sampling frame, selecting a sampling method, determining sample size and implementing the plan. It provides formulas to calculate sample sizes for proportions and means based on confidence level, precision and population characteristics. The goal is to obtain a representative sample and minimize error.
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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 sa ...
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Confidence Interval ModuleOne of the key concepts of statist.docx
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.
Estimating population values ppt @ bec doms
This document discusses confidence intervals for estimating population parameters. It covers confidence intervals for the mean when the population standard deviation is known and unknown, as well as confidence intervals for the population proportion. Key points include:
- A confidence interval provides a range of plausible values for an unknown population parameter based on a sample statistic.
- The margin of error and confidence level affect the width of a confidence interval.
- The t-distribution is used instead of the normal when the population standard deviation is unknown.
- Sample size formulas allow determining the required sample size to estimate a population parameter within a specified margin of error and confidence level.
Standard Error & Confidence Intervals.pptx
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
Confidence Intervals
This document discusses confidence intervals for population means and proportions. It explains how to construct confidence intervals using the normal distribution for large sample sizes (n ≥ 30) and the t-distribution for small sample sizes. Formulas are provided for calculating margin of error and determining necessary sample size. Guidelines are given for determining whether to use the normal or t-distribution based on sample size and characteristics. Confidence intervals can be constructed for variance and standard deviation using the chi-square distribution.
How to compute for sample size.pptx
To calculate the required sample size for a survey, you need to determine the population size, margin of error, confidence level, and standard deviation. The formula used is sample size = (z-score)2 * p(1-p) / (margin of error)2. For a survey estimating smartphone ownership in a city with a 99% confidence level, 0.03 margin of error, and 0.70 estimated standard deviation, the required sample size is 2,400.
Inferential statistics
The document discusses key concepts in statistical inference including estimation, confidence intervals, hypothesis testing, and types of errors. It provides examples and formulas for estimating population means from sample data, calculating confidence intervals, stating the null and alternative hypotheses, and making decisions to accept or reject the null hypothesis based on a significance level.
lecture13MWF.pdflecture13MWF.pdflecture13MWF.pdf
This document discusses confidence intervals and margin of error in statistical analysis. It defines key terminology like population mean, sample mean, standard deviation, and standard error. It explains that the margin of error depends on the sample size and standard deviation, and provides the formula for calculating sample size needed to achieve a given margin of error. Several examples are provided to illustrate how to construct confidence intervals and determine required sample sizes.
Chapter 6 part1- Introduction to Inference-Estimating with Confidence (Introd...
Introduction to Inference, Estimating with Confidence, Inference, Statistical Confidence, Confidence Intervals, Confidence Interval for a Population Mean, Choosing the Sample Size
Lect 10 Sample Size Estimation.ppt
- The document discusses factors to consider when determining sample size for quantitative studies, including desired level of confidence, margin of error, and how dispersed the population is.
- Sample size is a balance between what is desirable and feasible, and too small a sample may miss important effects or be too imprecise, while too large wastes resources.
- Formulas are provided to calculate the minimum required sample size for estimating a single population proportion based on the expected proportion, desired margin of error, and confidence level. Adjustments may be needed if the population is small.
Section 7 Analyzing our Marketing Test, Survey Results .docx
Section 7
Analyzing our Marketing Test, Survey Results
and Other Metrics Using Confidence Intervals
Rhonda Knehans Drake
Associate Professor, New York University
Data Analytics, Interpretation and Reporting
copyright © 2013
2
• When we estimate population averages or percentages based on
samples, a certain amount of error is present.
• The amount of error present is a function mostly of your
sample size.
• The larger the sample, the less error in your estimates.
• Today we will learn how to place bounds around our estimates
obtained.
• With such an interval, we will then be able to say with 90%, 95%
or 99% confidence that the true population estimate will lie in
these bounds.
Introduction
3
The eight confidence interval formulas we will discuss are for
the following situations:
1. Average or mean based on large samples
2. Average or mean based on small samples
3. Response rate or survey percent based on large samples
4. Response rate or survey percent based on small samples
5. Difference between 2 averages for large samples
6. Difference between 2 averages for small samples
7. Difference between 2 response rates for large samples
8. Difference between 2 response rates for small sample
Confidence Intervals
A-B Split Tests
4
1. Confidence Interval for Averages or Means Based on
Large Samples (n ≥ 30)
To calculate a confidence interval around a mean, the following
information is required:
– The sample mean x obtained from the test.
– The sample standard deviation S obtained from the test.
Many software packages, including Microsoft ExcelTM, can
automatically calculate this value for you. (Review Section 3 for the
standard deviation formula.)
– The sample size n of the test.
This is the number of observations used to calculate your mean. The
sample size must be greater than or equal to 30 in size.
– The desired confidence level: 90%, 95% or 99%.
A confidence interval constructed around the mean will guarantee, with
your specified level of confidence, the true mean will fall within those
bounds.
Sample Means
(Large Sample)
5
• Once all information is known, you construct the confidence
interval around the mean by adding and subtracting from the
mean a multiple of your standard deviation associated with the
sample mean. The “multiple” depends on your desired
confidence level.
• The formula for a confidence interval around the mean is
calculated as follows:
x (Z )(S/n )
Where:
• S is the standard deviation associated with the sample.
• n is the sample size.
• Z is equal to 1.645, 1.96 or 2.575 for a 90%, 95% or 99%
confidence level
Sample Means
(Large Sample)
6
Example:
Money magazine conducts a survey of 100 retirees across the US
and asks them how much they have in their retirement fund.
You obtain an average of $84.75 with a standard deviation of
$18.75, both in thousands of doll.
Statistik 1 7 estimasi & ci
This document outlines key concepts related to estimation and confidence intervals. It defines point estimates as single values used to estimate population parameters and interval estimates as ranges of values within which the population parameter is expected to occur. Confidence intervals provide an interval range based on sample observations within which the population parameter is expected to fall at a specified confidence level, such as 95% or 99%. The document discusses how to construct confidence intervals for the population mean when the population standard deviation is known or unknown.
Sample size
This document explains how to calculate sample size for a survey or poll. It outlines the key factors needed: population size, margin of error, confidence level, and standard deviation. It provides the formula for calculating sample size and works through an example using a 95% confidence level, 0.5 standard deviation, and +/- 5% margin of error. Based on these parameters, the sample size needed is 385 respondents. The document advises adjusting the confidence level or margin of error if the calculated sample size is too large.
Sampling Theory.pdf
The document discusses various sampling techniques for research. It defines key concepts like population, sample, sampling frame and sampling methods. It explains that the sampling process involves defining the target population, choosing a sampling frame, selecting a sampling method, determining sample size and implementing the plan. It provides formulas to calculate sample sizes for proportions and means based on confidence level, precision and population characteristics. The goal is to obtain a representative sample and minimize error.
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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 sa ...
Lecture_Wk08.pdf
- The document provides an overview and refresher on key statistical concepts like p-values, alpha levels, z-scores, t-scores, standard errors, and how to perform t-tests. It uses an example of calcium levels in milk bottles to demonstrate these concepts.
- Key points covered include how p-values and alpha levels are used to reject or fail to reject the null hypothesis, how the normal and t-distributions are used, and how standard errors and sample sizes impact results. Examples are worked through to show how to calculate z-scores, t-scores, and interpret the results of t-tests.
101_sampling__population_Sept_2020.ppt
The document discusses key concepts in statistics related to populations, samples, and sampling distributions. Some main points:
- We collect sample data to make inferences about unknown population parameters. Samples should be representative of the overall population.
- The sampling distribution of sample means approximates a normal distribution as long as sample sizes are large. This allows us to calculate confidence intervals and test hypotheses about population means and proportions.
- Common statistical tests include z-tests and t-tests for single means and proportions using the standard error to determine confidence intervals and assess significance. These can determine if sample results align with hypothesized population values.
Bbs11 ppt ch08
This chapter discusses confidence interval estimation for means and proportions. It introduces key concepts such as point estimates, confidence intervals, and confidence levels. For a mean where the population standard deviation is known, the confidence interval formula uses the normal distribution. When the standard deviation is unknown, the t-distribution is used instead. For a proportion, the confidence interval adds an allowance for uncertainty to the sample proportion. The chapter also covers determining sample sizes and interpreting confidence intervals.
Sample Size Determination
This document discusses key concepts related to determining sample size for surveys:
- Confidence interval and confidence level describe the level of certainty or precision in a sample - a 95% confidence level means the true population value would fall within the confidence interval 95% of the time.
- Sample size, population size, and response distribution (how answers are split) all impact the required sample size to achieve a given confidence level and interval. Higher confidence or lower intervals require larger samples.
- For a population of 20,000, with a 50-50 response split, and 95% confidence level, the required sample size is 377 people.
Presentation_advance_1n.pptx
This document provides an introduction to uncertainty in measurements. It discusses types of uncertainties including type A and type B uncertainties, standard uncertainty, confidence intervals, and expanded uncertainty. It explains key concepts such as degrees of freedom, standard deviation, standard error, t-distributions, and how to calculate confidence intervals and combined uncertainties. Examples are provided to demonstrate how to evaluate uncertainties and calculate best estimates, standard uncertainties, coverage factors, and confidence intervals for measurements.
Normal and standard normal distribution
The document discusses normal and standard normal distributions. It provides examples of using a normal distribution to calculate probabilities related to bone mineral density test results. It shows how to find the probability of a z-score falling below or above certain values. It also explains how to determine the sample size needed to estimate an unknown population proportion within a given level of confidence.
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February 17 Blackboards
- 1. An interval covering the span from –z to +z, centred on the sample mean X-bar, will “catch” the population mean mu for 95% of possible sample means on the normal distribution curve. (In this case z=1.96)
- 3. Confidence interval formulae for mu where n>= 30 and sigma is known (top), and where n>= 30 but sigma is not known (bottom).
- 4. Confidence interval formulae for the mean when n<30 and sigma is known (top), and n<30 but sigma not known (bottom). The variable has to be normally distributed to use either of these – if it isn’t, we don’t know how to deal (in this course).
- 6. Confidence interval formula for sample proportion. Works as long as n is big enough that n times p-hat is >=5, and n times q-hat is also >=5.
- 7. Example of confidence interval for a proportion
- 9. The “margin of error” E is half the width of the confidence interval. This is what people are talking about when they say things like “plus or minus 5%, 19 times out of 20” (19/20=0.95, or 95% confidence)
- 10. If you know the level of confidence you want, and the margin of error you are willing to tolerate, you can calculate how much data you will need to collect (minimum sample size).
- 11. Example of calculating minimum sample size needed to estimate a proportion.
- 12. Example continued. Note you should ALWAYS round up (you have to take a whole extra person to get 0.1 of a person).
- 13. Example of finding a confidence interval for a variance or standard deviation.
- 15. Another example
- 16. End of example