Estimating the Difference between Two Population Proportions
Question 2 • CA-Statistics and Probability B
A local school board claims that there is a difference in the proportions of households with school-aged children that would support starting the school year a week earlier, and the proportion of households without school-aged children that would support starting the school year a week earlier. They survey a random sample of 40 households with school-aged children about whether they would support starting the school year a week earlier, and 30 households respond yes. They survey a random sample of 45 households that do not have school-aged children, and 25 respond yes.Based on the 90% confidence interval, (0.03, 0.36), is there convincing evidence of a difference in the true proportions of households, those with school-aged children and those without school-aged children, who would support starting school early?
Answer
A
There is convincing evidence because the two sample proportions are different.
B
There is convincing evidence because the entire interval is above 0.
C
There is not convincing evidence because if another interval with a higher confidence level is calculated, it might contain 0.
D
There is not convincing evidence because two different sample sizes were used. In order to determine a difference, the same number of households should be selected from each population.
Explanation
The 90% confidence interval for the difference in the true proportions is (0.03, 0.36). Because the entire interval is above 0, it supports a positive difference rather than no difference, so there is convincing evidence that the proportions differ. The sample proportions are 30/40 = 0.75 and 25/45 ≈ 0.56, consistent with that conclusion.
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