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Unit 5 Overview: Sampling Distributions

5 min readโ€ขdecember 31, 2022

Harrison Burnside

Harrison Burnside

Josh Argo

Josh Argo

Jed Quiaoit

Jed Quiaoit

Harrison Burnside

Harrison Burnside

Josh Argo

Josh Argo

Jed Quiaoit

Jed Quiaoit

"This unit applies to sampling, introducing students to sampling distributions of statistics they will use when performing in Units 6 and 7. Students should understand that can be used to estimate corresponding and that () and () for these sampling distributions can be determined directly from the when certain sampling criteria are met. For large enough samples from any population, these sampling distributions can be approximated by a . Simulating sampling distributions helps students to understand how the values of statistics vary in repeated from populations with known parameters." -- College Board, AP Statistics course description

What is a Sampling Distribution?

A is a distribution where we take ALL possible samples of a given size and put those together as a data set.

For example, let's say we are looking at average number of snap peas taken from a field. If we take all possible samples of size 30, average each field, and then average those averages together, we would get a REALLY good picture of what the population parameter was (which is likely unrealistic to actually calculate). Sampling distributions are important because they lead the way to statistical : the act of making a prediction or testing a claim regarding a population parameter.

https://cdn.pixabay.com/photo/2017/10/25/22/29/bayesian-2889576_960_720.png

image courtesy of: pixabay.com

Sampling Distribution for Proportions

The first type of you will encounter is a used to estimate a population proportion.

For a , we will take the sample proportion from all possible samples of our given size and average those together to find the of our . Our is found using a formula given on the reference page. Once you have those two things, you have the crux of a for population proportion.

Conditions for Sampling Distribution

As we get into statistical , you'll find that sampling distributions hinge on certain conditions that make our sampling distributions an accurate portrayal of our population proportion.

Random

The first and possibly most important condition necessary for creating a is that our sample is randomly selected. If our sample is not randomly selected, then all the math and calculations we do are all for naught because our , or sample statistic, is biased. ๐Ÿ˜ฑ

Independence (10% Condition)

In order for the formula to be accurate, our samples have to be chosen independently of one another. Since we are sampling without replacement, this is technically impossible. However, by checking the 10% condition, we can determine that the amount of dependence is so negligible that our samples are essentially independent.

In order to check this condition, you need to make sure that the population is at least 10 times our sample size! โœ…

Normality (Large Counts Condition)

In order to eventually calculate the probability of obtaining certain samples using a , we need to verify that our is approximately normal.

For categorical data (proportions), we need to check the , which states that the number of expected successes and failures are at least 10. In other words, np is greater than or equal to 10 and n(1-p) is greater than or equal to 10.

Sampling Distribution for Means

When dealing with means, our center is the average of all of our sample means from all possible samples of size n. In other words, it's the average of the averages. Our is found by dividing our population by the square root of our sample size. As our sample size increases, our decreases, which plays a huge part in why a large sample size is vital in accurately estimating our . ๐Ÿค“

Conditions for Sampling Distribution

As you will find as we get into statistical , sampling distributions hinge on certain conditions that make our sampling distributions an accurate portrayal of our population proportion.

Random

Just as with estimating population proportions, it is essential that our is based on random samples. No mathematics or fancy statistics can "fix" a biased sample. ๐Ÿ˜•

Independence (10% Condition)

Again, as with population proportions, we must check the 10% condition the same way as we do for population proportions

Normality (Central Limit Theorem)

Our check to be sure that our is normal is different than our condition for population proportions. In order to make sure the for our is normal, we must verify one of two things: either that our population is normally distributed or our sample size is at least 30. This is known as the .

Sampling Distributions for the Differences in Means and Proportions

The last type of we encounter is when we are seeing if there is a difference in two populations. In this type of , our center is the difference in our two samples (which is presumably 0 if the two populations are not different). The necessary formulas for the center and spread of these sampling distributions can be found on the reference page. This plays a huge part in statistical when checking if two populations are in fact different, which is essential in .

Conditions for Inference

In order to check the conditions for when there are two samples, you are basically doing the same checks above but doing it twice: checking randomness, independence, and normality for both samples. ๐Ÿก

๐ŸŽฅ Watch: AP Stats - Sampling Distributions for Means

Key Terms to Review (20)

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.

Experimental Studies

: Experimental studies are research designs in which the researcher manipulates an independent variable to observe its effect on a dependent variable, while controlling for other variables. They allow researchers to establish cause-and-effect relationships between variables.

Independence (10% Condition)

: The independence assumption, also known as the 10% condition, states that for a random sample to be considered independent, the sample size should be no more than 10% of the population.

Inference

: Inference involves drawing conclusions or making predictions about a population based on sample data. It allows us to make generalizations and statements about a larger group using information from a smaller subset.

Large Counts Condition

: The large counts condition, also known as the "success-failure" condition, is used when applying certain statistical methods to categorical data. It states that for these methods to be valid, both the number of successes and failures must be at least 10.

Mean

: The mean is the average of a set of numbers. It is found by adding up all the values and dividing by the total number of values.

Measures of Center

: Measures of center refer to statistical measures that represent the central tendency or average of a set of data. They provide a single value that summarizes the typical or central value within a dataset.

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.

Point Estimate

: A point estimate is a single value that is used to estimate an unknown population parameter. It is obtained from sample data and serves as our best guess for the true value of the parameter.

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.

Population Parameters

: Population parameters are numerical values that summarize and describe an entire population. They represent fixed characteristics or properties of the entire group, but are often unknown and estimated using sample statistics.

Probabilistic Reasoning

: Probabilistic reasoning refers to using probabilities to make predictions or draw conclusions based on uncertain information or data.

Random Sampling

: Random sampling is a method of selecting individuals from a population in such a way that every individual has an equal chance of being chosen. It helps to ensure that the sample represents the population accurately.

Sample Statistics

: Sample statistics are numerical values that summarize and describe a sample of data. They provide information about the characteristics of the sample, such as its central tendency or variability.

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.

Sampling Distribution for Means

: The sampling distribution for means refers to the probability distribution of all possible sample means from samples of a fixed size taken from a population. It helps us understand how much variability we can expect in our sample means.

Sampling Distribution for Proportions

: The sampling distribution for proportions is a theoretical distribution that shows all possible sample proportions that could be obtained from repeated random samples of the same size from a population. It provides information about the variability and characteristics of sample proportions.

Standard Deviation

: The standard deviation measures the average amount of variation or dispersion in a set of data. It tells us how spread out the values are from the mean.

Variability

: Variability refers to the spread or dispersion of data points in a dataset. It measures how much the values differ from each other.

Unit 5 Overview: Sampling Distributions

5 min readโ€ขdecember 31, 2022

Harrison Burnside

Harrison Burnside

Josh Argo

Josh Argo

Jed Quiaoit

Jed Quiaoit

Harrison Burnside

Harrison Burnside

Josh Argo

Josh Argo

Jed Quiaoit

Jed Quiaoit

"This unit applies to sampling, introducing students to sampling distributions of statistics they will use when performing in Units 6 and 7. Students should understand that can be used to estimate corresponding and that () and () for these sampling distributions can be determined directly from the when certain sampling criteria are met. For large enough samples from any population, these sampling distributions can be approximated by a . Simulating sampling distributions helps students to understand how the values of statistics vary in repeated from populations with known parameters." -- College Board, AP Statistics course description

What is a Sampling Distribution?

A is a distribution where we take ALL possible samples of a given size and put those together as a data set.

For example, let's say we are looking at average number of snap peas taken from a field. If we take all possible samples of size 30, average each field, and then average those averages together, we would get a REALLY good picture of what the population parameter was (which is likely unrealistic to actually calculate). Sampling distributions are important because they lead the way to statistical : the act of making a prediction or testing a claim regarding a population parameter.

https://cdn.pixabay.com/photo/2017/10/25/22/29/bayesian-2889576_960_720.png

image courtesy of: pixabay.com

Sampling Distribution for Proportions

The first type of you will encounter is a used to estimate a population proportion.

For a , we will take the sample proportion from all possible samples of our given size and average those together to find the of our . Our is found using a formula given on the reference page. Once you have those two things, you have the crux of a for population proportion.

Conditions for Sampling Distribution

As we get into statistical , you'll find that sampling distributions hinge on certain conditions that make our sampling distributions an accurate portrayal of our population proportion.

Random

The first and possibly most important condition necessary for creating a is that our sample is randomly selected. If our sample is not randomly selected, then all the math and calculations we do are all for naught because our , or sample statistic, is biased. ๐Ÿ˜ฑ

Independence (10% Condition)

In order for the formula to be accurate, our samples have to be chosen independently of one another. Since we are sampling without replacement, this is technically impossible. However, by checking the 10% condition, we can determine that the amount of dependence is so negligible that our samples are essentially independent.

In order to check this condition, you need to make sure that the population is at least 10 times our sample size! โœ…

Normality (Large Counts Condition)

In order to eventually calculate the probability of obtaining certain samples using a , we need to verify that our is approximately normal.

For categorical data (proportions), we need to check the , which states that the number of expected successes and failures are at least 10. In other words, np is greater than or equal to 10 and n(1-p) is greater than or equal to 10.

Sampling Distribution for Means

When dealing with means, our center is the average of all of our sample means from all possible samples of size n. In other words, it's the average of the averages. Our is found by dividing our population by the square root of our sample size. As our sample size increases, our decreases, which plays a huge part in why a large sample size is vital in accurately estimating our . ๐Ÿค“

Conditions for Sampling Distribution

As you will find as we get into statistical , sampling distributions hinge on certain conditions that make our sampling distributions an accurate portrayal of our population proportion.

Random

Just as with estimating population proportions, it is essential that our is based on random samples. No mathematics or fancy statistics can "fix" a biased sample. ๐Ÿ˜•

Independence (10% Condition)

Again, as with population proportions, we must check the 10% condition the same way as we do for population proportions

Normality (Central Limit Theorem)

Our check to be sure that our is normal is different than our condition for population proportions. In order to make sure the for our is normal, we must verify one of two things: either that our population is normally distributed or our sample size is at least 30. This is known as the .

Sampling Distributions for the Differences in Means and Proportions

The last type of we encounter is when we are seeing if there is a difference in two populations. In this type of , our center is the difference in our two samples (which is presumably 0 if the two populations are not different). The necessary formulas for the center and spread of these sampling distributions can be found on the reference page. This plays a huge part in statistical when checking if two populations are in fact different, which is essential in .

Conditions for Inference

In order to check the conditions for when there are two samples, you are basically doing the same checks above but doing it twice: checking randomness, independence, and normality for both samples. ๐Ÿก

๐ŸŽฅ Watch: AP Stats - Sampling Distributions for Means

Key Terms to Review (20)

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.

Experimental Studies

: Experimental studies are research designs in which the researcher manipulates an independent variable to observe its effect on a dependent variable, while controlling for other variables. They allow researchers to establish cause-and-effect relationships between variables.

Independence (10% Condition)

: The independence assumption, also known as the 10% condition, states that for a random sample to be considered independent, the sample size should be no more than 10% of the population.

Inference

: Inference involves drawing conclusions or making predictions about a population based on sample data. It allows us to make generalizations and statements about a larger group using information from a smaller subset.

Large Counts Condition

: The large counts condition, also known as the "success-failure" condition, is used when applying certain statistical methods to categorical data. It states that for these methods to be valid, both the number of successes and failures must be at least 10.

Mean

: The mean is the average of a set of numbers. It is found by adding up all the values and dividing by the total number of values.

Measures of Center

: Measures of center refer to statistical measures that represent the central tendency or average of a set of data. They provide a single value that summarizes the typical or central value within a dataset.

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.

Point Estimate

: A point estimate is a single value that is used to estimate an unknown population parameter. It is obtained from sample data and serves as our best guess for the true value of the parameter.

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.

Population Parameters

: Population parameters are numerical values that summarize and describe an entire population. They represent fixed characteristics or properties of the entire group, but are often unknown and estimated using sample statistics.

Probabilistic Reasoning

: Probabilistic reasoning refers to using probabilities to make predictions or draw conclusions based on uncertain information or data.

Random Sampling

: Random sampling is a method of selecting individuals from a population in such a way that every individual has an equal chance of being chosen. It helps to ensure that the sample represents the population accurately.

Sample Statistics

: Sample statistics are numerical values that summarize and describe a sample of data. They provide information about the characteristics of the sample, such as its central tendency or variability.

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.

Sampling Distribution for Means

: The sampling distribution for means refers to the probability distribution of all possible sample means from samples of a fixed size taken from a population. It helps us understand how much variability we can expect in our sample means.

Sampling Distribution for Proportions

: The sampling distribution for proportions is a theoretical distribution that shows all possible sample proportions that could be obtained from repeated random samples of the same size from a population. It provides information about the variability and characteristics of sample proportions.

Standard Deviation

: The standard deviation measures the average amount of variation or dispersion in a set of data. It tells us how spread out the values are from the mean.

Variability

: Variability refers to the spread or dispersion of data points in a dataset. It measures how much the values differ from each other.


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ยฉ 2024 Fiveable Inc. All rights reserved.

APยฎ and SATยฎ are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.