Question

In Exercises 4.50 and 4.51 , a randomization distribution is given for a hypothesis test, and shows what values of the sample statistic are likely to occur if the null hypothesis is true. Several possible values are given for a sample statistic. In each case, indicate whether seeing a sample statistic as extreme as the value given is (i) reasonably likely to occur when the null hypothesis is true, (ii) unusual but might occur occasionally when the null hypothesis is true, or (iii) extremely unlikely to ever occur when the null hvpothesis is true. Figure 4.13 (a) shows a randomization distribution for a hypothesis test with \(H_{0}: p=0.30\). Answer the question for these possible sample proportions: (a) \(\hat{p}=0.1\) (b) \(\hat{p}=0.35\) (c) \(\hat{p}=0.6\)

Short answer

(a) \(\hat{p}=0.1\) is (iii) extremely unlikely to ever occur when the null hypothesis is true. (b) \(\hat{p}=0.35\) is (ii) unusual but might occur occasionally when the null hypothesis is true. (c) \(\hat{p}=0.6\) is (iii) extremely unlikely to ever occur when the null hypothesis is true.

Step-by-step solution

Understanding the Hypothesis

The null hypothesis \(H_{0}: p=0.30\), proposes that the population proportion \(p\) equals 0.30.

Comparing \(\hat{p}=0.1\) with the Null Hypothesis

\(p=0.1\) is quite a distance from the null value of \(p=0.30\). Hence, getting a sample proportion as extreme as \(\hat{p}=0.1\) is not reasonably likely to occur when the null hypothesis is true. We can categorize the probability of occurrence as (iii) extremely unlikely to ever occur when the null hypothesis is true.

Comparing \(\hat{p}=0.35\) with the Null Hypothesis

Comparing \(p=0.35\) with the null hypothesis, it is closer but still not identical. Thus, it's unusual but might occur occasionally when the null hypothesis is true. We can categorize it as (ii) unusual but might occur occasionally when the null hypothesis is true.

Comparing \(\hat{p}=0.6\) with the Null Hypothesis

\(p=0.6\) is very far from the null hypothesis. This is not at all likely considering our null hypothesis. So, it is (iii) extremely unlikely to ever occur when the null hypothesis is true.