Step 1 :Step 1 Null Hypothesis (H0): \( p_m = p_w \) Alternative Hypothesis (H1): \( p_m \neq p_w \) This is a two-tailed test.
Step 2 :Step 2 Calculate pooled proportion:\[p_p = \frac{x_m + x_w}{n_m + n_w} = \frac{0.45 * 20 + 0.70 * 80}{20 + 80} = \frac{9 + 56}{100} = 0.65\] Calculate Standard error:\[SE = \sqrt{\frac{p_p * (1-p_p)}{n_m} \times \frac{p_p * (1-p_p)}{n_w}} = 0.0791\]
Step 3 :Step 3 Calculate the test statistic:\[z = \frac{(p_m - p_w) - 0}{SE} = \frac{0.45 - 0.70}{0.0791} = -3.16\] Find the p-value from z-table:\[p = 2 * (1-P(-3.16)) = 0.0016\] Since the p-value is less than 0.05, we reject the null hypothesis. Thus, the proportion of men who own cats is significantly different than women who own cats at the 0.05 significance level.