AP Statistics 3.3 C Calculate an Appropriate Confidence Interval for a Population Proportion Questions

Section
1

The appropriate procedure is a one-sample z-interval for the proportion of all teenagers in the United States who would respond that they use a video streaming service every day.

Essentially correct (E) if the response satisfies the following two components:
1. Identifies the appropriate procedure as a one-sample z-interval by name or formula
or by the calculations of the correct
confidence interval endpoint values
2. States that the parameter of interest is the population proportion
Partially correct (P) if the response satisfies only one of the two components.
Incorrect (I) if the response does not meet the
criteria for E or P.

Additional Notes:

- The response to component 2 concerning the statement of "population proportion" can be found in any of the three sections of part (a).

- Any notation used to represent sample proportion or population proportion should remain consistent throughout part (a).

Model Solution

Scoring

(a)
Section
2

This survey selected a random sample of 920 teenagers in the United States, which enables the interval to be generalized to the population of interest. This sample of 920 teenagers is less than 10\% of the total number of teenagers in the United States, which is required as sampling was conducted without replacement from a finite population. In addition, there were more than 10 successes and 10 failures as (920)(0.59)=542.8( or 543) responded that they use a streaming service daily and (920)(0.41)=377.2 (or 377) responded that they did not. Thus, the sample size is large enough to support the assumption that the sampling distribution of $\hat{p}$ is approximately normal.

Therefore, a 95\% confidence interval for the population proportion is given by
$$\hat{p} \pm z^{*} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=0.59 \pm 1.96 \sqrt{\frac{(0.59)(0.41)}{920}},$$ which is $0.59 \pm 0.032$, and the interval is (0.558, 0.622).

Essentially correct (E) if the response satisfies the following four components:
2. Indicates 920 is less than ten percent of all teenagers in the United States
3. Verifies that there are at least 10 successes and failures by calculating the following
values $n \hat{p}=(920)(0.59) \approx 542.8$ and
$n(1-\hat{p})=(920)(1-0.59) \approx 377.2$
4. Reports the values for a correct interval
consistent with the procedure stated in Section 1
Partially correct (P) if the response satisfies
either component 3 or component 4 and at least one of the other three components.
Incorrect (I) if the response does not meet the
criteria for E or P.

Additional Notes:

- Stating the large sample condition without verification is not sufficient for component 3.

- If the response includes an inappropriate check of conditions, such as n>30, then component 3 is not satisfied.

- Supporting work, showing formulas or calculations, is not required for component 4.

- If the interval values are correct, the use of a one-sample z procedure for proportion can be used to satisfy component 1 of Section 1.

- A response that uses the value of x=543 will result in an interval of (0.5584, 0.6219), and a response that uses the value of x=542 will result in an interval of (0.5573, 0.6209). These interval endpoint values may be used to satisfy component 4.

- If the response includes supporting work for calculating the confidence interval that displays a correct formula with correct values inserted for $\hat{p}, n$, and z, then component 4 is satisfied even if values for the endpoints of the confidence interval are not displayed or calculated incorrectly.

- A response that computes an interval in percentages rather than proportions may satisfy component 4 if the response correctly indicates the use of percentages, (55.8\%, 62.2\%).

- Minor errors or omissions when checking assumptions may be considered if holistic scoring is required.

Model Solution

Scoring

(a)
Section
3

We can be 95\% confident that the proportion of
all teenagers in the United States who would respond that they use a streaming service every
day is between 0.558 and 0.622.

Essentially correct (E) if the response satisfies the following two components:
1. Indicates 95\% confidence and interprets the
interval using words such as "we are 95\%
confident" or "with 95\% confidence" and
provides interval endpoint values consistent with calculations in Section 2
2. Conveys inference about a population
proportion in the proper context, i.e., "the
true proportion of U.S. teenagers," "the
population proportion of U.S. teenagers," or "the proportion of all U.S. teenagers"
Partially correct (P) if the response satisfies only one of the two components.
Incorrect (I) if the response does not meet the
criteria for E or P.

Additional Notes:

- For component 1, responses that use the same incorrect proportion interval endpoint values calculated in Section 2 satisfies this portion of component 1.

- Section 3 is scored I if the response includes unreasonable values for a proportion or percentage.

- Responses must be consistent with the use of the terms proportion and percentage in component 2. A response that refers to the population percentage must use percentage values to satisfy component 2, and a response that refers to the population proportion must use proportion values.

- Any interpretation of the level of the interval, whether correct or incorrect, is considered extraneous and does not satisfy component 1.

- Clear indication of an inference to the sample of 920 teenagers, rather than the population of teenagers, does not satisfy component 2.

- Section 3 is scored I if it implies that the population proportion is a random variable. For example, it indicates that there is a 95 percent chance that the population proportion is between 0.558 and 0.622.

- A response that implies that we have found the actual percentage of U.S. teenagers who respond that they use a streaming service (deterministic) does not satisfy component 1.