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In hypothesis testing, the null hypothesis is an essential component. It's a statement made for the purpose of the test, which is assumed to be true until evidence suggests otherwise. In the given exercise, the null hypothesis (\( H_0 \)) asserts that the proportion of women who were attacked by relatives is less than or equal to the proportion of men attacked by relatives.This hypothesis provides a baseline or a neutral position against which the alternative hypothesis is tested. In mathematical terms, we formulate it as:\[H_0: p_1 \leq p_2\]Where \( p_1 \) represents the proportion of women attacked by relatives and \( p_2 \) represents the proportion of men attacked by relatives. By stating this hypothesis, we are essentially assuming for the time being that there is no greater prevalence of attacks among women.The null hypothesis is central because statistical tests are designed to evaluate whether there's enough evidence to reject it. The notion is not to prove it true but to determine the probability of observing the test results, assuming the null hypothesis holds true.

The alternative hypothesis presents a contrasting claim to the null hypothesis and is what researchers typically aim to support. For the exercise, the alternative hypothesis (\( H_1 \)) asserts that the percentage of women attacked by relatives is greater than that of men.It stands as the statement researchers hope to find evidence for in the analysis. The hypothesis is articulated as:\[H_1: p_1 > p_2\]Here, \( p_1 \) again represents the proportion of women who are victims of attacks by relatives, while \( p_2 \) reflects the proportion for men. The alternative hypothesis is essential for directing the focus of the investigation.Should the data indicate enough evidence against the null hypothesis at the selected significance level, the alternative hypothesis gains support. In this scenario, finding a significant result in favor of \( H_1 \) would imply sufficient statistical evidence that more women, as a proportion, experience such attacks compared to men.

A critical value serves as a threshold to determine whether to reject the null hypothesis. It depends on the chosen significance level (\( \alpha \)) and the type of test being performed. For this exercise, the level of significance is \( 0.10 \), and we're dealing with a right-tailed test.To find the critical value, we refer to the standard normal distribution, often using a Z-table. Given that the significance level is \( 0.10 \), the critical value is approximately:\[z_c \approx 1.28\]This critical value acts as a cutoff point. If our computed test statistic exceeds this value, we have grounds to reject the null hypothesis. The critical value, therefore, delineates the boundary between the region where we would retain the null hypothesis and the area where we have enough evidence to support the alternative.It's crucial to understand that setting the critical value is context-dependent, influenced by the significance level, and the design of the hypothesis test.

The test statistic provides a standardized measure to compare against the critical value. For this exercise, it quantifies the difference between sample proportions, taking into account variation within the samples.In computing the test statistic, sample data is compared under the assumption that the null hypothesis holds true. Here is the formula used:\[z = \frac{\hat{p}_1 - \hat{p}_2}{SE}\]Where:

• \( \hat{p}_1 = 0.3 \) - Represents the sample proportion of women attacked by relatives.
• \( \hat{p}_2 = 0.12 \) - Stands for the sample proportion of men attacked by relatives.
• \( SE = 0.0774 \) - This is the standard error, calculated as part of assessing variability within the sample.

Upon calculation, the test statistic is found to be \( z = 2.32 \). Since \( z = 2.32 \) is greater than the critical value of \( 1.28 \), we reject the null hypothesis. The test statistic thus plays an instrumental role, guiding the decision-making about the hypotheses.Ultimately, the test statistic translates the observed data's deviation from the null hypothesis into a number that allows us to establish the statistical significance of the results.