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5.7 Sampling Distributions for Sample Means

4 min readjanuary 2, 2023

Josh Argo

Josh Argo

B

Brianna Bukowski

Jed Quiaoit

Jed Quiaoit

Josh Argo

Josh Argo

B

Brianna Bukowski

Jed Quiaoit

Jed Quiaoit

Formulas

For these problems you are given a . The average of sample means is equal to the given mean. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-auLdfcFF50Gi.JPG?alt=media&token=e5d53fcd-3d66-4d04-a63d-e0e1d431d250

Source: AP Statistics Formula Sheet

For the standard deviation, when you deal with sample statistics, you must use standard error (SE). This version of standard deviation, which is used for population proportions in this case, is another comparison that must be memorized just like the chart in section 5.1.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-1U2MGu3frYMX.JPG?alt=media&token=dd332b46-065f-4c84-95d9-9755e8e89b76

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-7PHFviJGztAL.JPG?alt=media&token=1ce462bd-6a38-49a8-aba9-1cfa3cf018df

Source: The AP Statistics CED

Normal Condition: Central Limit Theorem

When you are working with numerical data and you want to estimate the mean of a population, you can use the of the (x̄) to make inferences about the . However, before you can solve the problem, you must first assure the is normally distributed using the . 😄

One important property of the of the is that it is approximately normal, provided the sample size is large enough. This means that even if the population distribution is not normal, the of the can be modeled using a if the sample size is large enough.

The "large enough" sample size is often taken to be 30 or greater. This is known as the , which states that the of the becomes approximately normal as the sample size increases, regardless of the shape of the population distribution. ⬭

Practice Problem

Suppose that you are conducting a study to estimate the average income of small business owners in your state. You decide to use a of 100 small business owners, and you collect data on their annual incomes. After analyzing the data, you find that the income is $50,000 per year with a standard deviation of $10,000. 💰

a) Explain what the for the represents and why it is useful in this situation.

b) Suppose that the true income for small business owners in your state is actually $45,000 per year. Describe the shape, center, and spread of the for the in this case.

c) Explain why the applies to the for the in this situation.

d) Discuss one potential source of that could affect the results of this study, and explain how it could influence the estimate of the income.

Answer

a) The for the represents the distribution of possible values for the if the study were repeated many times. It is useful in this situation because it allows us to make inferences about the based on the sample data.

b) If the true income for small business owners in your state is $45,000 per year, the for the would be approximately normal with a center at $45,000 and a spread that depends on the sample size and the variability of the population.

c) The applies to the for the in this situation because the sample size (n = 100) is large enough for the distribution to be approximately normal, even if the population is not normally distributed.

d) One potential source of in this study could be , which occurs when certain groups of individuals are more or less likely to be included in the sample. For example, if the sample is drawn from a list of small business owners who have registered with a particular organization, it could be biased toward those who are more financially successful or who are more likely to be active members of the organization. This could lead to an overestimate of the income for small business owners in the state.

On the other hand, if the sample is drawn from a list of small business owners who have applied for a particular loan program, it could be biased toward those who are less financially successful or who are more likely to be in need of financial assistance. This could lead to an underestimate of the income.

Key Terms to Review (9)

Bias

: Bias refers to a systematic deviation from the true value or an unfair influence that affects the results of statistical analysis. It can occur during data collection, sampling, or analysis and leads to inaccurate or misleading conclusions.

Central Limit Theorem

: The Central Limit Theorem states that as the sample size increases, the sampling distribution of the mean approaches a normal distribution regardless of the shape of the population distribution.

Normal Distribution

: A normal distribution is a symmetric bell-shaped probability distribution characterized by its mean and standard deviation. It follows a specific mathematical formula called Gaussian distribution.

Population Mean

: The population mean is the average value of a variable for an entire population. It represents a summary measure for all individuals or units within that population.

Sample Mean

: The sample mean is the average of a set of observations or data points taken from a larger population.

Sampling Distribution

: A sampling distribution refers to the distribution of a statistic (such as mean, proportion, or difference) calculated from multiple random samples taken from the same population. It provides information about how sample statistics vary from sample to sample.

Selection Bias

: Selection bias occurs when certain individuals or groups are more likely to be included or excluded from a study sample, leading to distorted results that may not represent the target population accurately.

Simple Random Sample

: A simple random sample is a subset of individuals from a larger population, where each individual has an equal chance of being selected. It ensures that every member of the population has an equal opportunity to be included in the sample.

Standard Error (SE)

: The standard error measures how much variability or spread there is in sample means that would be obtained from repeated sampling from the same population.

5.7 Sampling Distributions for Sample Means

4 min readjanuary 2, 2023

Josh Argo

Josh Argo

B

Brianna Bukowski

Jed Quiaoit

Jed Quiaoit

Josh Argo

Josh Argo

B

Brianna Bukowski

Jed Quiaoit

Jed Quiaoit

Formulas

For these problems you are given a . The average of sample means is equal to the given mean. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-auLdfcFF50Gi.JPG?alt=media&token=e5d53fcd-3d66-4d04-a63d-e0e1d431d250

Source: AP Statistics Formula Sheet

For the standard deviation, when you deal with sample statistics, you must use standard error (SE). This version of standard deviation, which is used for population proportions in this case, is another comparison that must be memorized just like the chart in section 5.1.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-1U2MGu3frYMX.JPG?alt=media&token=dd332b46-065f-4c84-95d9-9755e8e89b76

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-7PHFviJGztAL.JPG?alt=media&token=1ce462bd-6a38-49a8-aba9-1cfa3cf018df

Source: The AP Statistics CED

Normal Condition: Central Limit Theorem

When you are working with numerical data and you want to estimate the mean of a population, you can use the of the (x̄) to make inferences about the . However, before you can solve the problem, you must first assure the is normally distributed using the . 😄

One important property of the of the is that it is approximately normal, provided the sample size is large enough. This means that even if the population distribution is not normal, the of the can be modeled using a if the sample size is large enough.

The "large enough" sample size is often taken to be 30 or greater. This is known as the , which states that the of the becomes approximately normal as the sample size increases, regardless of the shape of the population distribution. ⬭

Practice Problem

Suppose that you are conducting a study to estimate the average income of small business owners in your state. You decide to use a of 100 small business owners, and you collect data on their annual incomes. After analyzing the data, you find that the income is $50,000 per year with a standard deviation of $10,000. 💰

a) Explain what the for the represents and why it is useful in this situation.

b) Suppose that the true income for small business owners in your state is actually $45,000 per year. Describe the shape, center, and spread of the for the in this case.

c) Explain why the applies to the for the in this situation.

d) Discuss one potential source of that could affect the results of this study, and explain how it could influence the estimate of the income.

Answer

a) The for the represents the distribution of possible values for the if the study were repeated many times. It is useful in this situation because it allows us to make inferences about the based on the sample data.

b) If the true income for small business owners in your state is $45,000 per year, the for the would be approximately normal with a center at $45,000 and a spread that depends on the sample size and the variability of the population.

c) The applies to the for the in this situation because the sample size (n = 100) is large enough for the distribution to be approximately normal, even if the population is not normally distributed.

d) One potential source of in this study could be , which occurs when certain groups of individuals are more or less likely to be included in the sample. For example, if the sample is drawn from a list of small business owners who have registered with a particular organization, it could be biased toward those who are more financially successful or who are more likely to be active members of the organization. This could lead to an overestimate of the income for small business owners in the state.

On the other hand, if the sample is drawn from a list of small business owners who have applied for a particular loan program, it could be biased toward those who are less financially successful or who are more likely to be in need of financial assistance. This could lead to an underestimate of the income.

Key Terms to Review (9)

Bias

: Bias refers to a systematic deviation from the true value or an unfair influence that affects the results of statistical analysis. It can occur during data collection, sampling, or analysis and leads to inaccurate or misleading conclusions.

Central Limit Theorem

: The Central Limit Theorem states that as the sample size increases, the sampling distribution of the mean approaches a normal distribution regardless of the shape of the population distribution.

Normal Distribution

: A normal distribution is a symmetric bell-shaped probability distribution characterized by its mean and standard deviation. It follows a specific mathematical formula called Gaussian distribution.

Population Mean

: The population mean is the average value of a variable for an entire population. It represents a summary measure for all individuals or units within that population.

Sample Mean

: The sample mean is the average of a set of observations or data points taken from a larger population.

Sampling Distribution

: A sampling distribution refers to the distribution of a statistic (such as mean, proportion, or difference) calculated from multiple random samples taken from the same population. It provides information about how sample statistics vary from sample to sample.

Selection Bias

: Selection bias occurs when certain individuals or groups are more likely to be included or excluded from a study sample, leading to distorted results that may not represent the target population accurately.

Simple Random Sample

: A simple random sample is a subset of individuals from a larger population, where each individual has an equal chance of being selected. It ensures that every member of the population has an equal opportunity to be included in the sample.

Standard Error (SE)

: The standard error measures how much variability or spread there is in sample means that would be obtained from repeated sampling from the same population.


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.