Step 1 :Define the null and alternative hypotheses. The null hypothesis is that the proportion of successful challenges for men (p1) is equal to the proportion of successful challenges for women (p2). The alternative hypothesis is that these proportions are not equal. Therefore, the hypotheses are: \[\begin{array}{l} H_{0}: p_{1}=p_{2} \\ H_{1}: p_{1} \neq p_{2} \end{array}\]
Step 2 :Calculate the test statistic and the P-value. The test statistic for a hypothesis test for the difference between two proportions is a z-score, which is calculated using the formula: \[z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}\] where \(\hat{p}_1\) and \(\hat{p}_2\) are the sample proportions, \(\hat{p}\) is the pooled sample proportion, and \(n_1\) and \(n_2\) are the sample sizes.
Step 3 :Calculate the sample proportions: \(p1 = 0.303202846975089\), \(p2 = 0.2796833773087071\), and the pooled sample proportion: \(p = 0.29496070272769304\).
Step 4 :Calculate the standard error: \(se = 0.020551533653407082\).
Step 5 :Calculate the z-score: \(z = 1.1444143324302583\).
Step 6 :Calculate the P-value: \(p_value = 0.252451847093357\).
Step 7 :The calculated z-score is approximately 1.14, which matches the given test statistic. The calculated P-value is approximately 0.252, which is much larger than the significance level of 0.01. This means that we do not have enough evidence to reject the null hypothesis that the proportions of successful challenges for men and women are equal.
Step 8 :Final Answer: The null and alternative hypotheses for the hypothesis test are: \[\begin{array}{l} H_{0}: p_{1}=p_{2} \\ H_{1}: p_{1} \neq p_{2} \end{array}\] The test statistic is \(z \approx 1.14\), and the P-value is \(P-value \approx 0.252\).