AP Statistics 4.5: Mean Test Conclusions

Step 1: States a correct pair of hypotheses.
Let $\mu_{\text {diff }}$ represent the population mean difference in purchase price (woman - man) for identically equipped cars of the same model, sold to both men and women by the same dealer, in the county.

The hypotheses to be tested are $\mathrm{H}_{0}: \mu_{\text {diff }}=0$ versus $\mathrm{H}_{\mathrm{a}}: \mu_{\text {diff }}>0$.
Step 2: Identifies a correct test procedure (by name or by formula) and checks appropriate conditions.
The appropriate procedure is a paired t-test.
The conditions for the paired t-test are:

The sample is randomly selected from the population.

The population of price differences (woman - man) is normally distributed, or the sample size is large.

The first condition is met because the car models and the individuals were randomly selected. The sample size (n=8) is not large, so we need to investigate whether it is reasonable to assume that the population of price differences is normally distributed. The dotplot of sample price differences reveals a fairly symmetric distribution, so we will consider the second condition to be met.

Step 3: Correct mechanics, including the value of the test statistic and p-value (or rejection region).
The test statistic is $t=\frac{585-0}{\frac{530.71}{\sqrt{8}}} \approx 3.12$.
The p-value, based on a t-distribution with 8-1=7 degrees of freedom, is 0.008.
Step 4: States a correct conclusion in the context of the study, using the result of the statistical test.
Because the p-value is very small (for instance, smaller than $\alpha=0.05$ ), we reject the null hypothesis. The data provide convincing evidence that, on average, women pay more than men in the country for the same car model.

Scoring

Each of steps 1, 2, 3, and 4 were scored as essentially correct (E), partially correct (P), or incorrect (I).
Step 1 is scored as follows:
Essentially correct (E) if the response identifies the correct parameter AND states correct hypotheses.
Partially correct (P) if the response identifies the correct parameter OR states correct hypotheses, but not both.

Incorrect (I) if the response does not meet the criteria for E or P.
Note: Defining the parameter symbol in context or simply using common parameter notation is sufficient.

Step 2 is scored as follows:
Essentially correct (E) if the response identifies the correct test procedure (by name or by formula) AND checks both conditions correctly.

Partially correct (P) if the response correctly completes two of the three components (identification of procedure, check of randomness condition, check of normality condition).

Incorrect (I) if the response does not meet the criteria for E or P.
Note: The random sampling condition can be verified by referring to the random selection of car models or to the random selection of male and female car buyers.

Step 3 is scored as follows:
Essentially correct (E) if the response correctly calculates both the test statistic and the p-value.
Partially correct (P) if the response correctly calculates the test statistic but not the p-value; OR if the response calculates the test statistic incorrectly but then calculates the correct p-value for the computed test statistic.

Incorrect (I) if the response does not meet the criteria for E or P.
Note: If the response identifies a z-test for a mean as the correct procedure in step 2, then the response can earn a P in step 3 if both the test statistic and the p-value are calculated correctly.

Step 4 is scored as follows:
Essentially correct (E) if the response provides a correct conclusion in context, also providing justification based on linkage between the p-value and the conclusion.

Partially correct (P) if the response provides a correct conclusion with linkage to the p-value, but not in context;

OR
if the response provides a correct conclusion in context, but without justification based on linkage to the p-value.

Incorrect (I) if the response does not meet the criteria for E or P.
Notes:

- If the conclusion is consistent with an incorrect p-value from step 3 and also in context with justification based on linkage to the p-value, step 4 is scored as E.

- A response that performs a two-sample t-test with correct calculations should fail to reject $\mathrm{H}_{0}$. A conclusion that is equivalent to "accept $\mathrm{H}_{0}$ " (such as "we conclude that women pay the same amount as men, on average"), either as a stated decision or as a conclusion in context, cannot be scored as E. Such a response will be scored as P provided that the conclusion is in context with linkage. Such a response will be scored as I if it lacks either context or linkage.

Each essentially correct (E) step counts as 1 point. Each partially correct (P) step counts as $\frac{1}{2}$ point.

Complete Response

Substantial Response

Developing Response

Minimal Response

If a response is between two scores (for example, $2 \frac{1}{2}$ points), use a holistic approach to decide whether to score up or down, depending on the overall strength of the response and communication.