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When conducting a statistical hypothesis test, the first step is to state the null hypothesis (H_0). This hypothesis proposes that there is no effect or difference in the population parameters. In our example, the null hypothesis states that the population proportions are equal: \[ H_0: p_1 = p_2 \] The null hypothesis serves as a baseline that we test against the alternative hypothesis. We either reject or fail to reject the null hypothesis based on the test results.

Next, we state the alternative hypothesis (H_a). This is the hypothesis that there is an effect or a difference. In our problem, the alternative hypothesis posits that the population proportions are not equal: \[ H_a: p_1 eq p_2 \] The alternative hypothesis is what we hope to provide evidence for through our test. It is often what the researcher wants to prove or validate.

The test statistic is a standardized value that we calculate from sample data during a hypothesis test. It allows us to determine how far our sample statistic is from the null hypothesis. In this exercise, we use the Z-test statistic for comparing population proportions. Here’s how it’s done: 1. Calculate the sample proportions: \[ \hat{p}_1 = \frac{x_1}{n_1} = \frac{28}{254} \approx 0.1102 \] \[ \hat{p}_2 = \frac{x_2}{n_2} = \frac{36}{301} \approx 0.1196 \] 2. Compute the pooled sample proportion: \[ \hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{28 + 36}{254 + 301} \approx 0.1152 \] 3. Find the standard error (SE): \[ SE = \sqrt{\hat{p}(1 - \hat{p})} \left( \frac{1}{n_1} + \frac{1}{n_2} \right) = \sqrt{(0.1152)(1 - 0.1152)} \left( \frac{1}{254} + \frac{1}{301} \right) \approx 0.0294 \] 4. Finally, calculate the test statistic (Z): \[ Z = \frac{\hat{p}_1 - \hat{p}_2}{SE} = \frac{0.1102 - 0.1196}{0.0294} \approx -0.32 \]

The critical value is a threshold that determines the boundary(s) of the rejection region(s). For our two-tailed test at the \( \alpha = 0.05 \) level of significance, the critical values are: \[ Z = \pm 1.96 \] If our test statistic falls beyond these critical values (in the rejection region), we reject the null hypothesis. In this exercise, the test statistic ( \( Z = -0.32 \)) does not fall outside the range of the critical values. Hence, we do not reject the null hypothesis.

The P-value helps us understand the probability of obtaining our test results, or something more extreme, assuming the null hypothesis is true. For our test statistic ( \( Z = -0.32 \)), we determine the P-value using the standard normal distribution table. The P-value for \( Z = -0.32 \) is approximately 0.7486. Since this is a two-tailed test, we multiply by 2: \[ P = 2 \times 0.7486 \approx 0.4972 \] The P-value represents the strength of the evidence against the null hypothesis. A smaller P-value indicates strong evidence against the null hypothesis, while a larger P-value suggests weak evidence.

The standard error (SE) measures the accuracy with which a sample distribution represents a population by using sample statistics. It is crucial when computing the test statistic. For example, the standard error in our problem is calculated as follows: \[ SE = \sqrt{\hat{p}(1 - \hat{p})} \left( \frac{1}{n_1} + \frac{1}{n_2} \right) \] where \( \hat{p} \) is the pooled sample proportion, giving us an SE of approximately 0.0294. The SE helps us understand how much variability there is in our sample.

Population proportions are a parameter of interest in many hypothesis tests. They represent the fraction of the population that has a particular characteristic. In our test, we compare two population proportions (\( p_1 \) and \( p_2 \)). The hypotheses are concerned with whether these proportions are the same or different. Sampling helps us estimate these population proportions, and we use these estimates to conduct our hypothesis tests. Understanding how these proportions work and how they can be compared is essential for various statistical analyses.