AP Statistics 3.3 Proportion Intervals

A polling agency showed the following two statements to a random sample of 1,048 adults in the United States.

Environment statement: Protection of the environment should be given priority over economic growth.
Economy statement: Economic growth should be given priority over protection of the environment.

The order in which the statements were shown was randomly selected for each person in the sample. After reading the statements, each person was asked to choose the statement that was most consistent with his or her opinion. The results are shown in the table.

Table for Question 3.3.A—Identify an appropriate confidence interval procedure including the parameter for a population proportion question 1 — AP Statistics

A suggestion was made to use a two-sample z-interval for a difference between proportions to investigate whether the difference in proportions between adults in the United States who would have chosen the environment statement and adults in the United States who would have chosen the economy statement is statistically significant. Is the two-sample z-interval for a difference between proportions an appropriate procedure to investigate the difference? Justify your answer.

Part (c):

The suggested procedure is not appropriate because one of the requirements for using a two-sample z-interval for a difference between proportions is that the two proportions are based on two independent samples. In the situation described the two proportions come from a single sample and thus are not independent.

Scoring

This question is scored in three sections. Section 1 consists of the mechanics of the confidence interval in part (a), section 2 consists of interpreting the confidence interval in part (a), and section 3 consists of parts (b) and (c).

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

States the correct procedure by name or formula

Calculates the confidence interval

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

Notes:

- A formula with correct values is sufficient for component 2.

- Component 2 can never be satisfied if the response contains an incorrect formula.

Section 2 is scored as follows:
Essentially correct (E) if the response includes the following four components for the interval interpretation:

Estimates a proportion

Infers about the population

States 95 percent confidence

Includes context

Partially correct (P) if the response satisfies components 1 and 2 AND satisfies only one of components 3 or 4;

OR
if the response gives a correct interpretation of the confidence level in context without interpreting the specific interval.

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

- Any indication of an inference to the adults sampled rather than the population does not satisfy component 2 and is scored as I.

- Stating values that are unrealistic as proportions (including blanks) in the interpretation lowers the score by one level (from E to P or from P to I).

- When both the interpretation of the interval and the level are given, only the interpretation of the interval is scored.

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

Part (b) states that the condition implies the sampling distribution of $\hat{p}$ is approximately normal OR the normal approximation to the binomial distribution is appropriate.

Part (c) indicates the procedure is not appropriate because the two proportions come from a single sample (dependent) rather than two (independent) samples.

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

- Referring to the sampling distribution needing to be normal without explicitly stating $\hat{p}$ satisfies component 1.

- Referring to normal approximation must specify "to binomial" to satisfy component 1.

- Discussing the two-sample z-interval versus the two-proportion z-interval is extraneous information.

- In part (c) component 2 can never be satisfied if a procedure that is not possible for one sample of categorical data is suggested.

Complete Response
Three parts essentially correct

Substantial Response
Two parts essentially correct and one part partially correct

Developing Response
Two parts essentially correct and no parts partially correct

OR
One part essentially correct and one or two parts partially correct

OR
Three parts partially correct

Minimal Response
One part essentially correct

OR
No parts essentially correct and two parts partially correct

Intent of Question

The primary goals of this question were to assess a student's ability to (1) use a scatterplot to comment on a report about the relationship between two variables and interpret the slope for the least-squares regression line summarizing this relationship; (2) describe the relationship between two variables in a scatterplot when a categorical variable is introduced and compare a characteristic of the distribution of a variable for different categories of individuals in a scatterplot; and (3) describe how the associations between two variables for each category of individuals in a scatterplot differ from the overall association in the same scatterplot.

Solution

3.3.A—Identify an appropriate confidence interval procedure including the parameter for a population proportion question 1