Independent random samples of 100 luxury cars and 250 non-luxury cars in a certain city are examined to see if they have bumper stickers. Of the 250 non-luxury cars, 125 have bumper stickers and of the 100 luxury cars, 30 have bumper stickers. Which of the following can be used to estimate the difference in the proportion of non-luxury cars with bumper stickers and the proportion of luxury cars with bumper stickers?

Question

Independent random samples of 100 luxury cars and 250 non-luxury cars in a certain city are examined to see if they have bumper stickers.  Of the 250 non-luxury cars, 125 have bumper stickers and of the 100 luxury cars, 30 have bumper stickers.  Which of the following can be used to estimate the difference in the proportion of non-luxury cars with bumper stickers and the proportion of luxury cars with bumper stickers?

BlackTom AI · Accepted Answer

The scenario describes two independent samples drawn from two distinct groups (luxury vs non-luxury cars), and we are interested in the difference between their proportions that have bumper stickers. This calls for a confidence interval that estimates a difference between two proportions, using the standard normal (z) approach because the samples are reasonably large (n1 = 100, n2 = 250) and the success/failure counts are adequate for a normal approximation.

Option 1: Two sample z confidence interval for a difference in two proportions. This aligns with the setup: two independent samples, estimating p1 - p2 with a z-based interval. We use pooled or unpooled variance depending on the method, but the key is that it’s a difference between two proportions from independent samples, making this the appropriate choice.

Option 2: One sample z confidence interval for a population mean. This would be used when estimating a mean from a single sample, not a proportion or a difference between two proportions. The data here are counts of a binary outcome (bumper sticker: yes/no), not a single mean.

Option 3: Paired t confidence interval for a mean difference. This applies when measurements are paired or matched (e.g., before-and-after on the same subjects) and the goal is a mean difference. Our problem involves two independent samples, so pairing does not apply.

Option 4: Two sample t confidence interval for a difference in two means. This is used for comparing the difference between two means from two independent samples when the outcome is continuous and the population variances are unknown. Here, the outcome is a proportion, not a mean, making this inappropriate.

Option 5: One sample z confidence interval for a population proportion. This would estimate a single proportion from one sample, not the difference between two proportions from two samples. Since we want p1 − p2, this single-proportion interval does not capture the comparison between groups.

In summary, option 1 correctly matches the problem’s structure: independent samples and a target of the difference between two proportions, using a two-sample z-based confidence interval.

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

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