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7.4 Setting Up a Test for a Population Mean

7 min readjanuary 4, 2023

Jed Quiaoit

Jed Quiaoit

Josh Argo

Josh Argo

Jed Quiaoit

Jed Quiaoit

Josh Argo

Josh Argo

When given a statistical claim from an article or previous study, the first necessary thing to do is to identify the test needed. Some key phrases you will see that tells us that a is called for is "do the data give convincing evidence..." or "is there convincing evidence of..." 🤔

When one of these key phrases appear in the prompt, we then need to determine if our data is categorical data or quantitative data. If we have quantitative data, we will set up a test for a . As with confidence intervals, a test for will make use of . If we only have one sample, we will perform a .

A one-sample t-test is used to compare the mean of a sample to a known . It is often used when the standard deviation (σ) of the population is not known. To conduct a one-sample t-test, you first need to determine the null and alternative hypotheses. The is a statement of no difference or no effect, and it is the hypothesis that is being tested, while the is the opposite of the , and it is what you hope to show is true. 1️⃣

Significance Level

One major aspect of our significance test is the . The (alpha, 𝞪) is the probability of rejecting the when it is actually true. It is the threshold that you set for determining whether the sample mean is significantly different from the claimed . If the p-value is less than the , then you can reject the and conclude that the sample mean is significantly different from the claimed . 🔝

It's important to choose an appropriate because setting it too low can lead to a high rate of false positives (also known as Type I errors), where you reject the when it is actually true. On the other hand, setting it too high can lead to a high rate of false negatives (also known as Type II errors), where you fail to reject the when it is actually false.

The most common used in research is 0.05, which means that there is a 5% chance of rejecting the when it is actually true. This is often considered a good balance between minimizing the risk of Type I and Type II errors. However, the appropriate will depend on the specific research question and the context in which the research is being conducted.

Connection to Confidence Interval

A is directly connected to a one sample t-interval. If we have a of 0.05, we would be looking at a sample statistic that would not occur in our 95% . If we select a of 0.02, it matches with a 98% . In summation, the complement of our will match with the confidence level of the matching . 🌉

Example

In the image below, we see a 95% in the non-shaded region for a mean of 0 and 29 df. The shaded region is known as the . This is the region in which a statistic is significant enough (in accordance with its 𝞪 level) to reject the claimed . In this example, any sample statistic greater than 2.04 or less than -2.04 would lead us to doubt that the true is 0. 🙅

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-09%20at%203.20.58%20PM.png?alt=media&token=56346a2e-400e-43bf-9408-881a2b4f63c9

image created with: https://homepage.divms.uiowa.edu/mbognar/

Writing Hypotheses

Once we have identified our test and , we need to identify and write our hypothesized values. We have two hypotheses: null and alternate.

Null Hypothesis (Ho)

In a one-sample t-test, the is a statement about the , and it is based on the claim made by the previous study or article. The is typically stated as 𝞵 = μ0, where 𝞵 is the sample mean and μ0 is the . 🥚

For example, if the previous study claimed that the mean number of chicken nuggets per bag is 20, the for a one-sample t-test would be 𝞵 = 20. This means that the is that the true mean number of chicken nuggets per bag is equal to 20.

Our is always going to be 𝞵 = (in number format).

Down the road, the purpose of the one-sample t-test is to determine whether the sample mean is significantly different from the . If the p-value is less than the predetermined (usually 0.05), then you can reject the and conclude that the sample mean is significantly different from the . If the p-value is greater than the , then you cannot reject the , and you must accept the claim as being true.

Alternate Hypothesis (Ha)

The is the opposite of the , and it is what you hope to show is true. In a one-sample t-test, the can take one of three forms: 𝞵 ≠ μ0, 𝞵 < μ0, or 𝞵 > μ0, where 𝞵 is the sample mean and μ0 is the from the . 🐤

For example, if the is 𝞵 = 20 (the is 20), the could be 𝞵 ≠ 20 (the sample mean is not equal to 20), 𝞵 < 20 (the sample mean is less than 20), or 𝞵 > 20 (the sample mean is greater than 20).

The that you choose will depend on the specific research question and the context in which the research is being conducted. For example, if you are testing the claim that the mean number of chicken nuggets per bag is 20, and you expect the actual mean to be higher, you would choose the 𝞵 > 20. On the other hand, if you expect the actual mean to be lower, you would choose the 𝞵 < 20.

Example

A recent study has found that the average number of school days missed by a high school senior is 5.2 days. After taking a random sample of 150 high school seniors, our sample has an average of 4.1 days missed with a standard deviation of 0.4. Do the data give convincing evidence that the average number of days missed by a high school senior is less than the claim from the study?

  • Ho: 𝞵 = 5.2

  • Ha: 𝞵 < 5.2

For this example, our comes directly from the study. The comes from the fact that the question implies that we are checking to see if the actual value is less than the hypothesized value.

Checking Conditions

Once we have our test confirmed and our hypothesis developed, we need to check our conditions for inference to be sure that our test can accurately be carried out. Just as with confidence intervals, we have 3 conditions: 🧪

  • Random

  • Independent

  • Normal

Random

If we are planning on using our sample statistics to develop a statistical test, it is imperative that our sample was chosen randomly. This is important because we are planning on using our sample mean to draw inference or conclusions about our . 🍀

If our sample is not chosen randomly to mirror our population, we cannot make statistical claims about the given population.

Independence

Since we are more than likely sampling without replacement, our sample is not truly independent. However, if our sample is not super close to the population, the effect of sampling without replacement is said to be negligible enough that our sample is essentially independent. 🏁

To check that this condition is met, we must verify that it is reasonable to believe that our population is at least 10x that of our sample.

You should state, "It is reasonable to believe that there are ____ (10n) _________ (in context of our population)"

Normal

Since we will be using the normal curve in our next unit to actually calculate the values necessary to perform our test, we need to assure that our sampling distribution is approximately normal. 🔔

There are three options to check this:

  1. (sample size is at least 30)

  2. Population is given to be approximately normal.

  3. Distribution of sample data looks approximately symmetric with no apparent outliers or gaps. This can be shown with a quick, modified box-plot sketch of our sample data.

Checking these conditions in this order will be the least cumbersome attempt in verifying the normal condition. Only one is necessary to verify normality.

🎥 Watch: AP Stats - Inference: Hypothesis Tests for Means

Key Terms to Review (14)

Alternative Hypothesis

: The alternative hypothesis is a statement that contradicts or negates the null hypothesis. It suggests that there is a significant relationship or difference between variables.

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.

Confidence Interval

: A confidence interval is a range of values that is likely to contain the true value of a population parameter. It provides an estimate along with a level of confidence about how accurate the estimate is.

Hypothesized Population Mean

: The hypothesized population mean is the value that we assume to be true for a population parameter before conducting a statistical test. It serves as a starting point for hypothesis testing.

Null Hypothesis

: The null hypothesis is a statement of no effect or relationship between variables in a statistical analysis. It assumes that any observed differences or associations are due to random chance.

One Sample T-test

: A one-sample t-test is a statistical test used to determine whether there is enough evidence to conclude that a sample comes from a particular population with known or hypothesized mean.

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.

Random Independent Normal Conditions

: Random independent normal conditions refer to assumptions made when using certain statistical tests, such as confidence intervals and hypothesis tests. These conditions include random sampling, independence of observations, and approximately normal distribution.

Rejection region

: The rejection region represents the set of sample outcomes that lead us to reject the null hypothesis in favor of an alternative hypothesis. It consists of extreme sample results that are unlikely to occur if the null hypothesis were true.

Significance Level

: The significance level, also known as alpha (α), determines how much evidence we need to reject the null hypothesis. It represents the probability of making a Type I error.

Standard Deviation (σ)

: The standard deviation is a measure of how spread out the values in a data set are from the mean. It quantifies the average amount by which each value differs from the mean.

Statistical Significance Test

: A statistical significance test is a method used to determine if the results of a study or experiment are statistically significant, meaning that they are unlikely to have occurred by chance.

T-Scores

: T-scores are standardized scores that represent how many standard deviations an individual data point is away from the mean, using the sample standard deviation.

Type II error

: Type II error occurs when we fail to reject a null hypothesis that is actually false. In other words, it's the mistake of accepting the null hypothesis when we should have rejected it.

7.4 Setting Up a Test for a Population Mean

7 min readjanuary 4, 2023

Jed Quiaoit

Jed Quiaoit

Josh Argo

Josh Argo

Jed Quiaoit

Jed Quiaoit

Josh Argo

Josh Argo

When given a statistical claim from an article or previous study, the first necessary thing to do is to identify the test needed. Some key phrases you will see that tells us that a is called for is "do the data give convincing evidence..." or "is there convincing evidence of..." 🤔

When one of these key phrases appear in the prompt, we then need to determine if our data is categorical data or quantitative data. If we have quantitative data, we will set up a test for a . As with confidence intervals, a test for will make use of . If we only have one sample, we will perform a .

A one-sample t-test is used to compare the mean of a sample to a known . It is often used when the standard deviation (σ) of the population is not known. To conduct a one-sample t-test, you first need to determine the null and alternative hypotheses. The is a statement of no difference or no effect, and it is the hypothesis that is being tested, while the is the opposite of the , and it is what you hope to show is true. 1️⃣

Significance Level

One major aspect of our significance test is the . The (alpha, 𝞪) is the probability of rejecting the when it is actually true. It is the threshold that you set for determining whether the sample mean is significantly different from the claimed . If the p-value is less than the , then you can reject the and conclude that the sample mean is significantly different from the claimed . 🔝

It's important to choose an appropriate because setting it too low can lead to a high rate of false positives (also known as Type I errors), where you reject the when it is actually true. On the other hand, setting it too high can lead to a high rate of false negatives (also known as Type II errors), where you fail to reject the when it is actually false.

The most common used in research is 0.05, which means that there is a 5% chance of rejecting the when it is actually true. This is often considered a good balance between minimizing the risk of Type I and Type II errors. However, the appropriate will depend on the specific research question and the context in which the research is being conducted.

Connection to Confidence Interval

A is directly connected to a one sample t-interval. If we have a of 0.05, we would be looking at a sample statistic that would not occur in our 95% . If we select a of 0.02, it matches with a 98% . In summation, the complement of our will match with the confidence level of the matching . 🌉

Example

In the image below, we see a 95% in the non-shaded region for a mean of 0 and 29 df. The shaded region is known as the . This is the region in which a statistic is significant enough (in accordance with its 𝞪 level) to reject the claimed . In this example, any sample statistic greater than 2.04 or less than -2.04 would lead us to doubt that the true is 0. 🙅

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-09%20at%203.20.58%20PM.png?alt=media&token=56346a2e-400e-43bf-9408-881a2b4f63c9

image created with: https://homepage.divms.uiowa.edu/mbognar/

Writing Hypotheses

Once we have identified our test and , we need to identify and write our hypothesized values. We have two hypotheses: null and alternate.

Null Hypothesis (Ho)

In a one-sample t-test, the is a statement about the , and it is based on the claim made by the previous study or article. The is typically stated as 𝞵 = μ0, where 𝞵 is the sample mean and μ0 is the . 🥚

For example, if the previous study claimed that the mean number of chicken nuggets per bag is 20, the for a one-sample t-test would be 𝞵 = 20. This means that the is that the true mean number of chicken nuggets per bag is equal to 20.

Our is always going to be 𝞵 = (in number format).

Down the road, the purpose of the one-sample t-test is to determine whether the sample mean is significantly different from the . If the p-value is less than the predetermined (usually 0.05), then you can reject the and conclude that the sample mean is significantly different from the . If the p-value is greater than the , then you cannot reject the , and you must accept the claim as being true.

Alternate Hypothesis (Ha)

The is the opposite of the , and it is what you hope to show is true. In a one-sample t-test, the can take one of three forms: 𝞵 ≠ μ0, 𝞵 < μ0, or 𝞵 > μ0, where 𝞵 is the sample mean and μ0 is the from the . 🐤

For example, if the is 𝞵 = 20 (the is 20), the could be 𝞵 ≠ 20 (the sample mean is not equal to 20), 𝞵 < 20 (the sample mean is less than 20), or 𝞵 > 20 (the sample mean is greater than 20).

The that you choose will depend on the specific research question and the context in which the research is being conducted. For example, if you are testing the claim that the mean number of chicken nuggets per bag is 20, and you expect the actual mean to be higher, you would choose the 𝞵 > 20. On the other hand, if you expect the actual mean to be lower, you would choose the 𝞵 < 20.

Example

A recent study has found that the average number of school days missed by a high school senior is 5.2 days. After taking a random sample of 150 high school seniors, our sample has an average of 4.1 days missed with a standard deviation of 0.4. Do the data give convincing evidence that the average number of days missed by a high school senior is less than the claim from the study?

  • Ho: 𝞵 = 5.2

  • Ha: 𝞵 < 5.2

For this example, our comes directly from the study. The comes from the fact that the question implies that we are checking to see if the actual value is less than the hypothesized value.

Checking Conditions

Once we have our test confirmed and our hypothesis developed, we need to check our conditions for inference to be sure that our test can accurately be carried out. Just as with confidence intervals, we have 3 conditions: 🧪

  • Random

  • Independent

  • Normal

Random

If we are planning on using our sample statistics to develop a statistical test, it is imperative that our sample was chosen randomly. This is important because we are planning on using our sample mean to draw inference or conclusions about our . 🍀

If our sample is not chosen randomly to mirror our population, we cannot make statistical claims about the given population.

Independence

Since we are more than likely sampling without replacement, our sample is not truly independent. However, if our sample is not super close to the population, the effect of sampling without replacement is said to be negligible enough that our sample is essentially independent. 🏁

To check that this condition is met, we must verify that it is reasonable to believe that our population is at least 10x that of our sample.

You should state, "It is reasonable to believe that there are ____ (10n) _________ (in context of our population)"

Normal

Since we will be using the normal curve in our next unit to actually calculate the values necessary to perform our test, we need to assure that our sampling distribution is approximately normal. 🔔

There are three options to check this:

  1. (sample size is at least 30)

  2. Population is given to be approximately normal.

  3. Distribution of sample data looks approximately symmetric with no apparent outliers or gaps. This can be shown with a quick, modified box-plot sketch of our sample data.

Checking these conditions in this order will be the least cumbersome attempt in verifying the normal condition. Only one is necessary to verify normality.

🎥 Watch: AP Stats - Inference: Hypothesis Tests for Means

Key Terms to Review (14)

Alternative Hypothesis

: The alternative hypothesis is a statement that contradicts or negates the null hypothesis. It suggests that there is a significant relationship or difference between variables.

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.

Confidence Interval

: A confidence interval is a range of values that is likely to contain the true value of a population parameter. It provides an estimate along with a level of confidence about how accurate the estimate is.

Hypothesized Population Mean

: The hypothesized population mean is the value that we assume to be true for a population parameter before conducting a statistical test. It serves as a starting point for hypothesis testing.

Null Hypothesis

: The null hypothesis is a statement of no effect or relationship between variables in a statistical analysis. It assumes that any observed differences or associations are due to random chance.

One Sample T-test

: A one-sample t-test is a statistical test used to determine whether there is enough evidence to conclude that a sample comes from a particular population with known or hypothesized mean.

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.

Random Independent Normal Conditions

: Random independent normal conditions refer to assumptions made when using certain statistical tests, such as confidence intervals and hypothesis tests. These conditions include random sampling, independence of observations, and approximately normal distribution.

Rejection region

: The rejection region represents the set of sample outcomes that lead us to reject the null hypothesis in favor of an alternative hypothesis. It consists of extreme sample results that are unlikely to occur if the null hypothesis were true.

Significance Level

: The significance level, also known as alpha (α), determines how much evidence we need to reject the null hypothesis. It represents the probability of making a Type I error.

Standard Deviation (σ)

: The standard deviation is a measure of how spread out the values in a data set are from the mean. It quantifies the average amount by which each value differs from the mean.

Statistical Significance Test

: A statistical significance test is a method used to determine if the results of a study or experiment are statistically significant, meaning that they are unlikely to have occurred by chance.

T-Scores

: T-scores are standardized scores that represent how many standard deviations an individual data point is away from the mean, using the sample standard deviation.

Type II error

: Type II error occurs when we fail to reject a null hypothesis that is actually false. In other words, it's the mistake of accepting the null hypothesis when we should have rejected it.


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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.