In the confidence interval for p_1 - p_2, with hat p _1 = 62/100 = 0.62 and hat p _2 = 69/150 = 0.46, the estimated standard error SE(hat p _1 - hat p _2) equals:

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SSC Statistics QUESTION #2007

Question 1

In the confidence interval for \(p_1 - p_2\), with \(\hat{p}_1 = 62/100 = 0.62\) and \(\hat{p}_2 = 69/150 = 0.46\), the estimated standard error \(SE(\hat{p}_1 - \hat{p}_2)\) equals:

\(\sqrt{\dfrac{0.62(0.38)}{100} + \dfrac{0.46(0.54)}{150}}\)✔️
\(\sqrt{\dfrac{0.62(0.38) + 0.46(0.54)}{250}}\)
\(\dfrac{0.62 - 0.46}{\sqrt{250}}\)
\(\sqrt{\dfrac{0.54 \times 0.46}{250}}\)

Correct Answer Explanation

For independent samples and a confidence interval (not a hypothesis test), the SE is \(\sqrt{\dfrac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \dfrac{\hat{p}_2(1-\hat{p}_2)}{n_2}} = \sqrt{\dfrac{0.62 \times 0.38}{100} + \dfrac{0.46 \times 0.54}{150}}\). The pooled estimate is used only for hypothesis testing, not CI construction.

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