Three parts crammed into one question, which I did not expect.
Start by computing the joint probability using independence, then derive the expected count for neither. For the hypothesis test, clearly state the null and alternative hypotheses, choose a two-proportion z-test, and outline the steps including checking assumptions and interpreting the p-value.
Pro tip: When discussing hypothesis testing, emphasize the importance of considering practical significance and potential confounding factors, not just statistical significance. Also, mention that in real-world A/B tests, you often need to account for multiple comparisons and sequential testing.
Since honest and relevant are independent, P(honest and relevant) = P(honest) * P(relevant) = 0.7 * 0.8 = 0.56.
P(neither) = 1 - P(honest or relevant) = 1 - (P(honest) + P(relevant) - P(both)) = 1 - (0.7 + 0.8 - 0.56) = 1 - 0.94 = 0.06. So, out of 1000 answers, expected neither = 0.06 * 1000 = 60.
Define null hypothesis H0: p1 = p2 (relevance rates are equal) and alternative H1: p1 ≠ p2 (two-tailed). Choose significance level α = 0.05.
Calculate sample proportions, pooled proportion, standard error, and z-statistic. Find the p-value and compare to α. If p-value < 0.05, reject H0; otherwise, fail to reject H0.
Discuss the conclusion in context, including effect size and confidence intervals. Mention any assumptions (e.g., random sampling, independence) and potential limitations.
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