AP Statistics 3.8 Test Errors

According to a 2017 national survey in Country B, the mean number of bedrooms in newly built houses was 2.9. Rodney, a researcher, believes the mean number of bedrooms in newly built houses in the country was different in 2024 than it was in 2017. To investigate his belief, he took a large random sample of newly built houses in Country B in 2024 and recorded the number of bedrooms in each house. The distribution of the number of bedrooms for the sampled houses is summarized in the table.

Distribution of the Number of Bedrooms for the Houses Sampled in 2024

Rodney will use a one-sample t-test for a population mean to test his belief.

Explain, in context, what a Type I error would be for Rodney's hypothesis test.

[Maximum number: 1]

Suppose you did 10 independent tests of the form $H_{0}: \mu=25$ versus $H_{\mathrm{a}}: \mu<25$ , each at the $\alpha=0.05$ significance level. What is the probability of committing a Type I error and incorrectly rejecting a true $H_{0}$ with at least one of the 10 tests?

A

0.05

B

0.40

C

0.60

D

0.95

[Maximum number: 4]

Systolic blood pressure is the amount of pressure that blood exerts on blood vessels while the heart is beating. The mean systolic blood pressure for people in the United States is reported to be 122 millimeters of mercury (mmHg) with a standard deviation of 15 mmHg.
The wellness department of a large corporation is investigating whether the mean systolic blood pressure of its employees is greater than the reported national mean. A random sample of 100 employees will be selected, the systolic blood pressure of each employee in the sample will be measured, and the sample mean will be calculated.
Let $\mu$ represent the mean systolic blood pressure of all employees at the corporation. Consider the following hypotheses.

\begin{aligned} & \mathrm{H}_{0}: \mu=122 \\ & \mathrm{H}_{\mathrm{a}}: \mu>122 \end{aligned}

Scoring

This question is scored in three sections. Section 1 consists of part (a) and part (b), section 2 consists of part (c), and section 3 consists of part (d) and part (e). Sections 1, 2, and 3 are scored as essentially correct (E), partially correct (P), or incorrect (I).

Section 1 is scored as follows:
Essentially correct (E) if the response satisfies the following four components:

Part (a) includes in the description of a Type II error the fact that the alternative hypothesis is true, either generically or in context of the situation.

Part (a) includes in the description of a Type II error the fact that the null hypothesis is not rejected, either generically or in context of the situation.

Part (b) includes a correct z-score for the upper 5 percent tail and indicates the correct direction for the rejection region.

Part (b) includes $\mu_{\bar{x}}=122, \sigma_{\bar{x}}=1.5$ , and the resulting $\bar{x}$ value.

Partially correct (P) if the response satisfies only two or three of the four components.
Incorrect (I) if the response does not meet the criteria for E or P.
Notes:

- If the response in part (a) does not include context, the number of components satisfied is reduced by one (that is, from four to three, or from three to two, and so on). Context includes a reference to units, to blood pressure, to employees, etc.

- If a response in part (a) is clearly referring to an individual's blood pressure as opposed to the mean blood pressure of all employees, neither components 1 nor 2 are satisfied.

Section 2 is scored as follows:
Essentially correct (E) if the response satisfies the following three components:

Recognizes that the null hypothesis will be rejected when $\bar{x} \geq 124.4675$ , as found in part (b).

Provides the correct sampling distribution for the sample mean when the true mean is 125, including correct values for the mean and standard deviation, either explicitly or by plugging them into the test statistic formula.

Provides evidence of using the normal curve and finds the correct probability value.

Partially correct (P) if the response satisfies only two of the three components.
Incorrect (I) if the response does not meet the criteria for E or P.
Notes:

- Components 1 and 3 can still be satisfied if errors made in finding the rejection region in part (b) are carried into part (c).

- A calculator statement that does not include labels for input values does not satisfy component 2 but may still satisfy components 1 and 3.

Section 3 is scored as follows:
Essentially correct (E) if the response satisfies the following four components:

Part (d) specifies power as the name of the probability.

Part (e) correctly states that the probability would be greater.

Part (e) correctly implies that the standard deviation of the sampling distribution decreases, either explicitly or by substituting values into a formula.

Part (e) indicates the minimum value of $\bar{x}$ for which the null hypothesis is rejected decreases, either explicitly or by substituting values into a formula.

Note: Component 4 can still be satisfied if a response indicates that a maximum value of $\bar{x}$ for which the null hypothesis is rejected increases if this direction is consistent with answers in parts (b) and (c).

Partially correct (P) if the response satisfies only two or three of the four components.
Incorrect (I) if the response does not meet the criteria for E or P.

Complete Response
Three sections essentially correct

Substantial Response
Two sections essentially correct and one section partially correct

Developing Response
Two sections essentially correct and no sections partially correct

OR
One section essentially correct and one or two sections partially correct

OR
Three sections partially correct

Minimal Response
One section essentially correct

OR
No sections essentially correct and two sections partially correct

OR
Sections 1 and 2 incorrect, and section 3 partially correct with exactly three of the four components satisfied

OR
Section 1 partially correct with exactly three of the four components satisfied, and sections 2 and 3 incorrect

(a)

Describe a Type II error in the context of the hypothesis test.

(b)

What statistical term is used for the probability found in part (c) ?

(c)

Suppose the size of the sample of employees to be selected is greater than 100. Would the probability of rejecting the null hypothesis be greater than, less than, or equal to the probability calculated in part (c) ? Explain your reasoning.

Part (e):
If the sample size is increased from 100 to something larger, the probability of rejecting the null hypothesis when the population mean is 125 will be higher than it is for a sample of size 100. Intuitively, more data provide a higher probability of a correct conclusion. The technical explanation is that the rejection region will still be z>1.645, but the sampling distributions of the sample mean will have a smaller standard deviation;
therefore, the minimum value of $\bar{x}$ for which we would reject the null hypothesis would be lower and, in return, the probability the null hypothesis is rejected will increase.

AP Statistics 3.8 Potential Errors When Performing Tests Questions

Exam points

Covered syllabus points

Covered syllabus points

Question 1

Question 2

Question 3

Question (a)

Question (b)

Question (c)