Questions: Identify the acid and conjugate hypothesis. Let p1 be working men and p2 be socializing and communicating men. Choose the correct answer below. A. H0: p1 = p2 Ha: p1 > p2 B. H0: p1 = p2 Ha: p1 < p2 C. H0: p1 = p2 Ha: p1 ≠ p2 D. H0: p1 > p2 Ha: p1 ≤ p2 Round to two decimal places as needed. Percentage engaged in each activity: Women: 70% 59% Men: 35% 37%

Solution Steps

The question asks if the proportion of men who work is _greater_ than the proportion of men who socialize and communicate. Therefore, the alternative hypothesis (Ha) should reflect this proposed difference, and the null hypothesis (Ho) would state that there is no difference. Let $p_1$ represent the proportion of men working and $p_2$ represent the proportion of men socializing and communicating.

$H_0: p_1 \le p_2$

$H_a: p_1 > p_2$

The significance level is given as $\alpha = 0.10$. Since this is a one-tailed test (we are only interested if $p_1$ is _greater_ than $p_2$), we look up the z-score corresponding to $1 - \alpha$, or $1-0.10 = 0.90$ in a standard normal distribution table. This value is $z = 1.28$.

To calculate the standardized test statistic (z), we will use the following formula for the difference between two proportions:

$z = \frac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$

Where:

• $\hat{p}_1$ is the sample proportion of men who work. From the bar chart, this is $56\%$ or $0.56$.

• $\hat{p}_2$ is the sample proportion of men who socialize and communicate. From the bar chart, this is $37\%$ or $0.37$.

• $n_1$ is the sample size of men, which is 200.

• $n_2$ is also the sample size of men, also 200 (though it's the same group of men we compare the two proportions _within_ this group).

• $\hat{p}$ is the pooled proportion, calculated as: $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$, where $x_1$ and $x_2$ are the number of successes in each group. In this case, $x_1 = 0.56 * 200 = 112$ and $x_2 = 0.37 * 200 = 74$. Therefore, $\hat{p} = \frac{112 + 74}{200 + 200} = \frac{186}{400} = 0.465$

Under the null hypothesis, $(p_1 - p_2) = 0$. Plugging these values into the formula:

$z = \frac{(0.56 - 0.37) - 0}{\sqrt{0.465(1-0.465)(\frac{1}{200} + \frac{1}{200})}} = \frac{0.19}{\sqrt{0.465(0.535)(\frac{2}{200})}} \approx \frac{0.19}{0.0357} \approx 5.32$

Final Answer: