AP Statistics 4.5 Carrying Out a Test for a Population Mean or Population Mean Difference Study Notes - New Syllabus
AP Statistics 4.5 One-Sample t-Tests: Test Statistic, p-Value, and Conclusions Study Notes – New Syllabus
AP Statistics 4.5 One-Sample t-Tests: Test Statistic, p-Value, and Conclusions Study Notes – As per latest AP Statistics Syllabus.
LEARNING OBJECTIVES
- 4.5.A Calculate an appropriate test statistic and p-value for testing a hypothesis about a population mean or population mean difference.
- 4.5.B Interpret the p-value of a hypothesis test for a population mean or population mean difference.
- 4.5.C Justify a claim about the population based on the results of a hypothesis test for a population mean or population mean difference.
ESSENTIAL KNOWLEDGE:
- 4.5.A.1 The test statistic for a one-sample t-test for a population mean or population mean difference is
\( t=\dfrac{\bar{x}-\mu_0}{\dfrac{s}{\sqrt{n}}} \), where \(n\) has degrees of freedom \(n-1\). The t-statistic has a t-distribution with degrees of freedom \(n-1\) when the null hypothesis is true. - 4.5.A.2 The p-value for a one-sample t-test for a population mean or population mean difference is found using the appropriate t-distribution table or technology.
- 4.5.B.1 The p-value is the probability of obtaining a test statistic as extreme or more extreme than the test statistic that was observed (i.e., in the direction of the alternative hypothesis) given that the null hypothesis is true. An interpretation of the p-value of a hypothesis test for a population mean or population mean difference should include a statement that the p-value is computed by assuming that the null hypothesis is true (i.e., by assuming that the population mean is equal to the particular value stated in the null hypothesis in context).
- 4.5.C.1 A formal decision explicitly compares the p-value to the significance level, \( \alpha \). If the p-value < \( \alpha \), then reject the null hypothesis, \( H_0:\mu=\mu_0 \). If the p-value > \( \alpha \), then fail to reject the null hypothesis.
- 4.5.C.2 The results of a hypothesis test for a population mean or population mean difference can serve as the statistical reasoning to support the answer to an investigative question about the population that was sampled.
- 4.5.C.3 A conclusion for the hypothesis test for a population mean or population mean difference is stated in context consistent with, and in terms of, the alternative hypothesis using non-definitive language. The conclusion should contain a reference to the parameter and the population.
4.5.A.1 Test Statistic for a One-Sample t-Test
After verifying the conditions and stating the hypotheses, the next step in a one-sample t-test is to calculate the test statistic.
The test statistic measures how many standard errors the sample mean is above or below the hypothesized population mean.
If the null hypothesis is true, the test statistic follows a t-distribution with
\( n-1 \)
degrees of freedom.
Test Statistic Formula
\( t=\dfrac{\bar{x}-\mu_0}{\dfrac{s}{\sqrt{n}}} \)
Where:
- \( t \) = Test statistic
- \( \bar{x} \) = Sample mean
- \( \mu_0 \) = Hypothesized population mean from the null hypothesis
- \( s \) = Sample standard deviation
- \( n \) = Sample size
- \( \dfrac{s}{\sqrt{n}} \) = Standard error of the sample mean
Degrees of Freedom
\( df=n-1 \)
The degrees of freedom determine which t-distribution is used to calculate the p-value.
| Component | Meaning |
|---|---|
| \( \bar{x} \) | Sample mean |
| \( \mu_0 \) | Hypothesized population mean |
| \( s \) | Sample standard deviation |
| \( n \) | Sample size |
| \( df \) | Degrees of freedom = \(n-1\) |
Interpreting the Test Statistic
- A positive \(t\)-value indicates that the sample mean is greater than the hypothesized mean.
- A negative \(t\)-value indicates that the sample mean is less than the hypothesized mean.
- The farther the test statistic is from 0, the stronger the evidence against the null hypothesis.
Important AP Exam Notes
- Always use the hypothesized mean from the null hypothesis when calculating the test statistic.
- Use the sample standard deviation (\(s\)), not the population standard deviation.
- The degrees of freedom for a one-sample t-test are always
\( df=n-1 \)
- A larger absolute value of the test statistic generally leads to a smaller p-value.
Example
A company claims that the average battery life of its smartphones is 20 hours.
A random sample of 25 phones has
\( \bar{x}=18.8 \)
\( s=3 \)
Calculate the test statistic.
▶️ Answer / Explanation
Step 1: Write the formula.
\( t=\dfrac{\bar{x}-\mu_0}{\dfrac{s}{\sqrt{n}}} \)
Step 2: Substitute the given values.
\( t=\dfrac{18.8-20}{\dfrac{3}{\sqrt{25}}} \)
Step 3: Calculate the standard error.
\( \dfrac{3}{\sqrt{25}}=\dfrac{3}{5}=0.6 \)
Step 4: Calculate the test statistic.
\( t=\dfrac{-1.2}{0.6}=-2.00 \)
Step 5: Degrees of freedom.
\( df=25-1=24 \)
Answer:
The test statistic is
\( t=-2.00 \)
The negative value indicates that the sample mean is below the hypothesized population mean.
4.5.A.2 Finding the p-Value for a One-Sample t-Test
After calculating the test statistic, the next step is to determine the p-value.
The p-value is the probability of obtaining a test statistic that is as extreme as, or more extreme than, the observed value, assuming the null hypothesis is true.
For a one-sample t-test, the p-value is found using the appropriate t-distribution with
\( df=n-1 \)
degrees of freedom.
The p-value is typically obtained using a graphing calculator, statistical software, or a t-distribution table.
Calculator Steps (TI-84)
After calculating the test statistic:
STAT → TESTS → T-Test
Enter:
- \( \mu_0 \)
- \( \bar{x} \)
- \( s \)
- \( n \)
- Select the correct alternative hypothesis (\(<\), \(>\), or \(\ne\)).
The calculator reports both the test statistic and the p-value.
| Alternative Hypothesis | Area Used for the p-Value |
|---|---|
| \( H_a:\mu<\mu_0 \) | Left-tail area |
| \( H_a:\mu>\mu_0 \) | Right-tail area |
| \( H_a:\mu\ne\mu_0 \) | Both tails (two-tailed) |
Important AP Exam Notes
- The p-value is always calculated assuming the null hypothesis is true.
- The p-value is determined using the t-distribution with \(df=n-1\).
- The direction of the alternative hypothesis determines whether the test is left-tailed, right-tailed, or two-tailed.
- A smaller p-value provides stronger evidence against the null hypothesis.
Example
Using the battery-life data from the previous example, the calculated test statistic is \( t=-2.00 \) with \( df=24 \).
The alternative hypothesis is \( H_a:\mu<20 \).
Use technology to find the p-value.
▶️ Answer / Explanation
TI-84 Calculator Steps
STAT → TESTS → T-Test
Enter the sample statistics and select
\( H_a:\mu<20 \)
The calculator reports approximately
p-value ≈ 0.028
Interpretation
If the true mean battery life is 20 hours, there is approximately a 2.8% chance of obtaining a sample mean this low or lower due to random sampling variation alone.
4.5.B.1 Interpreting the p-Value
After calculating the test statistic, the next step in a hypothesis test is to interpret the p-value.
The p-value measures how likely it is to obtain a sample result that is as extreme as or more extreme than the observed result, assuming that the null hypothesis (\(H_0\)) is true.
In other words, the p-value tells us whether the observed sample result is unusual if the population parameter is actually equal to the value stated in the null hypothesis.
Definition of the p-Value
The p-value is the probability of obtaining a test statistic that is as extreme as or more extreme than the observed test statistic, in the direction of the alternative hypothesis, assuming the null hypothesis is true.
| Small p-Value | Large p-Value |
|---|---|
| The observed sample result is unusual if the null hypothesis is true. | The observed sample result is not unusual if the null hypothesis is true. |
| Provides stronger evidence against the null hypothesis. | Provides little or no evidence against the null hypothesis. |
General Interpretation Template
Assuming that the null hypothesis is true, the probability of obtaining a sample result (or test statistic) as extreme as or more extreme than the one observed is the p-value.
The interpretation should always be written in the context of the problem.
For a Population Mean
Assuming that the true population mean is equal to the value stated in the null hypothesis, the probability of obtaining a sample mean as extreme as or more extreme than the observed sample mean is the p-value.
For a Population Mean Difference
Assuming that the true population mean difference is equal to the value stated in the null hypothesis (usually \(0\)), the probability of obtaining a sample mean difference as extreme as or more extreme than the observed sample mean difference is the p-value.
Important AP Exam Notes
- Every p-value interpretation must begin by stating that the calculation is made assuming the null hypothesis is true.
- The p-value is a probability about the sample statistic, not the population parameter.
- The p-value is not the probability that the null hypothesis is true.
- A small p-value indicates that the observed result would be unusual if the null hypothesis were true.
- The interpretation must always include the context of the problem.
Example (Population Mean)
A company claims that the average battery life of its smartphones is 20 hours.
A one-sample t-test is performed with the hypotheses
\( H_0:\mu=20 \)
\( H_a:\mu<20 \)
The calculated p-value is
\( 0.028 \)
Interpret the p-value.
▶️ Answer / Explanation
Assuming that the true mean battery life of all smartphones is 20 hours, the probability of obtaining a sample mean as low as or lower than the one observed (or equivalently, a test statistic as small as or smaller than the observed value) is 0.028.
This means that there is a 2.8% chance of observing a sample result this extreme due to random sampling variation alone if the null hypothesis is true.
Example (Matched Pairs)
A researcher studies whether a tutoring program improves mathematics test scores.
The hypotheses are
\( H_0:\mu_d=0 \)
\( H_a:\mu_d>0 \)
where
Difference = After − Before
The calculated p-value is \( 0.012 \)
Interpret the p-value.
▶️ Answer / Explanation
Assuming that the true mean difference in mathematics test scores (After − Before) is 0, the probability of obtaining a sample mean difference as large as or larger than the one observed (or equivalently, a test statistic as large as or larger than the observed value) is 0.012.
This means that there is a 1.2% chance of obtaining a sample result this extreme due to random sampling variation alone if the null hypothesis is true.
4.5.C.1 Making a Decision Using the p-Value
After calculating the p-value, the next step in a hypothesis test is to make a formal statistical decision.
The decision is made by comparing the p-value with the significance level (\(\alpha\)).
The significance level is chosen before collecting the data and represents the maximum probability of rejecting a true null hypothesis (Type I Error) that the researcher is willing to accept.
Decision Rule
| Comparison | Decision |
|---|---|
| \( p\text{-value}\le\alpha \) | Reject the null hypothesis (\(H_0\)). |
| \( p\text{-value}>\alpha \) | Fail to reject the null hypothesis (\(H_0\)). |
Meaning of Each Decision
Reject the Null Hypothesis
- The sample provides convincing statistical evidence in favor of the alternative hypothesis.
- The observed sample result is unlikely to have occurred by random chance alone if the null hypothesis is true.
Fail to Reject the Null Hypothesis
- The sample does not provide convincing statistical evidence for the alternative hypothesis.
- This does not prove that the null hypothesis is true.
- It simply means that there is insufficient evidence to reject it.
Important AP Exam Notes
- The decision is based only on comparing the p-value with the significance level.
- Never write “Accept the null hypothesis.”
- Always write “Fail to reject the null hypothesis.”
- Rejecting the null hypothesis means there is convincing statistical evidence supporting the alternative hypothesis.
- Failing to reject the null hypothesis means there is not convincing statistical evidence supporting the alternative hypothesis.
Example 1
A researcher performs a one-sample t-test at the \( \alpha=0.05 \) significance level.
The calculated p-value is \( 0.032 \)
State the statistical decision.
▶️ Answer / Explanation
Compare the p-value with the significance level.
\(0.032<0.05\)
Since the p-value is less than the significance level, reject the null hypothesis.
There is convincing statistical evidence supporting the alternative hypothesis.
Example 2
A hypothesis test is conducted using \( \alpha=0.01 \)
The calculated p-value is \(0.084\)
State the statistical decision.
▶️ Answer / Explanation
Compare the p-value with the significance level.
\(0.084>0.01\)
Since the p-value is greater than the significance level, fail to reject the null hypothesis.
There is not convincing statistical evidence to support the alternative hypothesis.
4.5.C.2 Using the Results of a Hypothesis Test to Answer an Investigative Question
The purpose of a hypothesis test is to answer an investigative question about a population using evidence obtained from a sample.
- After making the statistical decision (reject or fail to reject the null hypothesis), the result should be used to answer the original research question.
- The conclusion is based on the sample evidence and is applied to the population from which the sample was drawn.
The hypothesis test provides statistical reasoning to determine whether the sample data provide convincing evidence for the claim stated in the alternative hypothesis.
Relationship Between the Hypothesis Test and the Investigative Question
| Decision | Answer to the Investigative Question |
|---|---|
| Reject \(H_0\) | The sample provides convincing statistical evidence to support the investigative claim. |
| Fail to Reject \(H_0\) | The sample does not provide convincing statistical evidence to support the investigative claim. |
Important AP Exam Notes
- The conclusion should answer the original investigative question.
- The conclusion must refer to the population, not only the sample.
- Statistical conclusions are based on sample evidence, not certainty.
- The hypothesis test determines whether the sample provides convincing evidence for the claim in the alternative hypothesis.
- Do not state that the hypothesis test proves a claim; it only provides evidence for or against it.
Example
A researcher wants to determine whether students who participate in an after-school tutoring program have a higher mean mathematics test score than the district average.
A one-sample t-test is performed, and the null hypothesis is rejected.
Use the result of the hypothesis test to answer the investigative question.
▶️ Answer / Explanation
Because the null hypothesis was rejected, the sample provides convincing statistical evidence that students who participate in the after-school tutoring program have a higher mean mathematics test score than the district average.
This conclusion answers the original investigative question and applies to the population represented by the sample.
4.5.C.3 Writing the Final Conclusion in Context
The final step of a hypothesis test is to write a conclusion that is consistent with the statistical decision and is stated in the context of the problem.
The conclusion should refer to:
- The population.
- The population parameter (\(\mu\) or \(\mu_d\)).
- The response variable.
- The claim stated in the alternative hypothesis.
The conclusion should use non-definitive language because statistical inference is based on sample evidence.
Correct Conclusion Templates
If \(H_0\) is Rejected
There is convincing statistical evidence that the true population mean (or population mean difference) is…
If \(H_0\) is Not Rejected
There is not convincing statistical evidence that the true population mean (or population mean difference) is…
| Decision | Correct Conclusion |
|---|---|
| Reject \(H_0\) | There is convincing statistical evidence to support the alternative hypothesis. |
| Fail to Reject \(H_0\) | There is not convincing statistical evidence to support the alternative hypothesis. |
Common AP Exam Mistakes
- Do not write “Accept the null hypothesis.”
- Do not say the alternative hypothesis has been proven.
- Do not say the population mean definitely equals or does not equal a value.
- Always include the population, the parameter, and the context.
- Use phrases such as “convincing statistical evidence” or “not convincing statistical evidence.”
Important AP Exam Notes
- The conclusion should always be consistent with the statistical decision.
- Reference the population parameter, not the sample statistic.
- Use cautious, non-definitive language because conclusions are based on sample evidence.
- The conclusion should answer the original research question in context.
Example
A one-sample t-test is conducted to determine whether the average amount of time that college students spend studying each week is greater than 15 hours.
The hypotheses are
\( H_0:\mu=15 \)
\( H_a:\mu>15 \)
The calculated p-value is \(0.018\)
At the 5% significance level, write the conclusion.
▶️ Answer / Explanation
Since
\(0.018<0.05\),
reject the null hypothesis.
There is convincing statistical evidence that the true mean number of hours studied each week by all college students is greater than 15 hours.
The conclusion refers to the population mean, is written in context, and uses appropriate non-definitive statistical language.