Question

suppose a researcher is testing the hypothesis $h_0: p = 0.6$ versus $h_1: p < 0.6$ and she finds the p - value to be 0.27. explain what this means. would she reject the null hypothesis? why? choose the correct explanation below. \\(\bigcirc\\) a. if the p - value for a particular test statistic is 0.27, she expects results no more extreme than the test statistic in about 27 of 100 samples if the null hypothesis is true. \\(\bigcirc\\) b. if the p - value for a particular test statistic is 0.27, she expects results at least as extreme as the test statistic in exactly 27 of 100 samples if the null hypothesis is true. \\(\bigcirc\\) c. if the p - value for a particular test statistic is 0.27, she expects results no more extreme than the test statistic in exactly 27 of 100 samples if the null hypothesis is true. \\(\bigcirc\\) d. if the p - value for a particular test statistic is 0.27, she expects results at least as extreme as the test statistic in about 27 of 100 samples if the null hypothesis is true.

Explanation:

The P - value is the probability of obtaining a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true. It is not an exact count (so "exactly" in options B and C is incorrect). Also, it's about results at least as extreme (not no more extreme, so option A is incorrect).

Answer:

D. If the P - value for a particular test statistic is 0.27, she expects results at least as extreme as the test statistic in about 27 of 100 samples if the null hypothesis is true.

Since the P - value (0.27) is greater than the common significance level (e.g., 0.05 or 0.10), she would not reject the null hypothesis. The decision rule is: reject \(H_0\) if \(P - value\leq\alpha\) (significance level). Here, \(0.27>0.05\) (assuming \(\alpha = 0.05\)).