Question

The null and alternative hypotheses for a test are given as well as some information about the actual sample(s) and the statistic that is computed for each randomization sample. Indicate where the randomization distribution will be centered. In addition, indicate whether the test is a left-tail test, a right-tail test, or a twotailed test. Hypotheses: \(H_{0}: p_{1}=p_{2}\) vs \(H_{a}: p_{1}>p_{2}\) Sample: \(\hat{p}_{1}=0.3, n_{1}=20\) and \(\hat{p}_{2}=0.167, n_{2}=12\) Randomization statistic \(=\hat{p}_{1}-\hat{p}_{2}\)

Short answer

The randomization distribution would be centered at 0, assuming the null hypothesis of no difference in proportions. This is a right-tail test because we are testing for the first proportion being greater than the second one.

Step-by-step solution

Decoding the Hypotheses

Given, null hypothesis (H₀): \(p_{1} = p_{2}\), i.e., the two populations have the same proportions. Alternative hypothesis (Ha): \(p_{1}> p_{2}\), i.e., the proportion of the first population is greater than the proportion of the second population.

Understanding Sample Data

The sample for population 1 has \(\hat{p}_{1} = 0.3\) and \(n_{1} = 20\), and for population 2 it has \(\hat{p}_{2} = 0.167\) and \(n_{2} = 12\). These values are estimates of population proportions and sample sizes for the two populations.

Center of Randomization Distribution

Under the null hypothesis, we assume that the two population proportions are equal. Hence, the randomization distribution of \(\hat{p}_{1}-\hat{p}_{2}\) will be centered at 0 as per the null hypothesis.

Test Type identification

As our alternative hypothesis says \(p_{1}> p_{2}\), we are interested in finding evidence of the first proportion being greater than the second one. This means our test is a right-tail test, because we are looking at the extreme right end of the distribution for evidence.