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AP Statistics 7.6 Confidence Intervals for the Difference of  Two Means- FRQs - Exam Style Questions

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Question

Independent random samples of \(500\) households were taken from a large metropolitan area in the United States for the years \(1950\) and \(2000\). Histograms of household size (number of people in a household) for the years are shown below.

(a) Compare the distributions of household size in the metropolitan area for the years \(1950\) and \(2000\).

(b) A researcher wants to use these data to construct a confidence interval to estimate the change in mean household size in the metropolitan area from the year \(1950\) to the year \(2000\). State the conditions for using a two-sample t-procedure, and explain whether the conditions for inference are met.

Most-appropriate topic codes (CED):

• TOPIC 1.9: Comparing Distributions of a Quantitative Variable — part (a)
• TOPIC 7.6: Confidence Intervals for the Difference of Two Means — part (b)
▶️ Answer/Explanation
Detailed solution

(a)
Center: Household size tended to be larger in \(1950\) than in \(2000\). The median household size was about \(5\) people in \(1950\) compared to \(3\)-\(4\) people in \(2000\).

Variability: The \(1950\) distribution showed more variability with standard deviation about \(2.6\) people and range of \(13\) people, compared to \(2000\) with standard deviation about \(2.1\) people and range of \(11\) people.

Shape: Both distributions are right-skewed. In both years, there are a few households with very large sizes (up to \(14\) people in \(1950\), \(12\) people in \(2000\)).

(b)
Conditions for two-sample t-procedure:
1. Independent random samples: Satisfied – independent random samples of \(500\) households each.
2. Normality/Large sample size: Satisfied – sample sizes of \(500\) each are large enough to overcome the right skewness (Central Limit Theorem applies).
3. Population size: Satisfied – metropolitan area population is much larger than \(10 \times 500 = 5,000\) households.

All conditions are met for using a two-sample t-procedure.

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