In a random sample of 60 shoppers chosen from the shoppers at a large suburban mall, 36 indicated that they had been to a movie in the past month. In an independent random sample of 50 shoppers chosen from the shoppers in a large downtown shopping area, 31 indicated that they had been to a movie in the past month. Which of the following inference procedures should be used to determine whether these data provide sufficient evidence to reject the hypothesis that the proportion of shoppers at the suburban mall who had been to a movie in the past month is the same as the proportion of shoppers in the large downtown shopping area who had been to a movie in the past month?

Question

In a random sample of 60 shoppers chosen from the shoppers at a large suburban mall, 36 indicated that they had been to a movie in the past month.  In an independent random sample of 50 shoppers chosen from the shoppers in a large downtown shopping area, 31 indicated that they had been to a movie in the past month.  Which of the following inference procedures should be used to determine whether these data provide sufficient evidence to reject the hypothesis that the proportion of shoppers at the suburban mall who had been to a movie in the past month is the same as the proportion of shoppers in the large downtown shopping area who had been to a movie in the past month?

BlackTom AI · Accepted Answer

The problem involves comparing proportions from two independent samples to assess whether the underlying population proportions are equal. Here is the analysis of each option:

Option 1: Paired t test for a mean difference
- This test is used when the data consist of paired observations (e.g., before-and-after measurements on the same subjects). Here, the samples are independent suburban mall shoppers and downtown shoppers, not paired. Therefore, a paired t test is inappropriate for comparing two independent proportions.

Option 2: Two sample t confidence interval for a difference two means
- This method constructs a confidence interval for the difference between two population means when the data are numerical and approximately normally distributed (or when sample sizes are large). Our outcome is a binary indicator (visited a movie in the past month), so the appropriate framework is proportions, not means. Thus, this option does not fit the data type or the hypotheses being tested.

Option 3: Two sample z test for a difference in two proportions
- This is the standard procedure for testing whether two independent population proportions are equal. It uses the pooled or unpooled proportion under the null hypothesis, depending on the exact test variant, and computes a z statistic for the difference between p1 and p2. Given our data are two independent samples with binary outcomes, this is the correct inference approach for testing H0: p1 = p2 against H1: p1 ≠ p2.

Option 4: Two sample t test for a difference in two means
- Similar to Option 2, this is appropriate for comparing two population means from numerical data. Since the variable is binary (movie attendance yes/no), this test is not suitable for proportions. Consequently, this option is inappropriate for the question.

Option 5: Two sample z confidence interval for a difference in two proportions
- While this option correctly uses a z-based framework for proportions and would yield a confidence interval for p1 − p2, the question asks which inference procedure should be used to determine whether there is evidence to reject the hypothesis p1 = p2. A z test (Option 3) is the corresponding inferential procedure for hypothesis testing. A confidence interval alone does not perform the hypothesis test, though it provides information about the plausible values of the difference.

In summary, the data involve two independent proportions from separate samples, so the appropriate inferential method is a two-sample z test for the difference in two proportions. Other options rely on paired data, means, or misapply confidence-interval concepts to a testing scenario, making them unsuitable for this question.

AP Statistics-Deputron-T2-2025 Trimester 2 Final Exam MC Review #1

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