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In hypothesis testing, the null hypothesis is a critical concept that serves as a starting point for statistical testing. It is often denoted as \( H_0 \) and represents a statement of no effect or no difference. In this exercise, the null hypothesis for the first scenario is \( \pi_1 \leq \pi_2 \). This implies that the proportion of success in the first population is less than or equal to that in the second population. The purpose of testing the null hypothesis is to assess whether there is enough evidence to reject it in favor of the alternative hypothesis, \( H_1 \), which posits that \( \pi_1 > \pi_2 \), implying that the first proportion is greater. Understanding the null hypothesis is crucial because it establishes a baseline or default position that the test seeks to falsify. If the data provide sufficient evidence against the null hypothesis, we conclude that the alternative hypothesis is more likely true. However, if the evidence is insufficient, we fail to reject the null hypothesis.

The test statistic is a standardized value used to determine whether to reject the null hypothesis. It offers a way to compare the sample data against what is expected under the null hypothesis. For a difference in proportions, the test statistic is calculated using the formula:\[Z = \frac{(\hat{p}_1 - \hat{p}_2) - D_0}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}\] Where:- \( \hat{p}_1 \) and \( \hat{p}_2 \) are the sample proportions.- \( D_0 \) is the hypothesized difference in proportions; in cases like this, it's often 0.- \( \hat{p} \) is the pooled proportion.The test statistic allows us to make comparisons based on standard normal distribution. In our example, the calculated \( Z \)-value is 1.6. We compare this value with a critical \( Z \)-value corresponding to the level of significance used, which is 0.05 for this test.

The pooled proportion \( \hat{p} \) is an essential element in hypothesis testing for two proportions. It combines the proportions from two samples into a single, unified estimate. This is necessary when testing the null hypothesis that assumes the proportions are equal. The pooled proportion is calculated using:\[ \hat{p} = \frac{X_1 + X_2}{n_1 + n_2} \]Where \( X_1 \) and \( X_2 \) are the number of successes in the first and second samples, respectively, and \( n_1 \) and \( n_2 \) are the sizes of the two samples. In the example, with \( X_1 = 70 \) and \( X_2 = 90 \), and sample sizes of 100 and 150:- \( \hat{p} = \frac{160}{250} = 0.64 \)This pooled estimate is used in calculating the standard error of the difference in proportions, thereby enabling the test statistic calculation.

The decision rule provides a guideline to decide when to reject the null hypothesis. It is based on the critical value obtained from the standard normal distribution, given a certain level of significance, often denoted by \( \alpha \). In this problem, with a significance level of 0.05, the decision is based on a right-tailed test with a critical value of approximately 1.645. - **Decision Rule**: Reject the null hypothesis if the test statistic \( Z \) is greater than 1.645.If \( Z \) is less than or equal to this critical value, we fail to reject the null hypothesis. In our example, the calculated test statistic is 1.6. Since this value does not exceed the critical value, the decision for the initial hypothesis is to fail to reject \( H_0: \pi_1 \leq \pi_2 \). This decision rule helps maintain the objectivity of the hypothesis testing process by providing a clear, quantitative threshold for decision-making.