To solve this problem, we'll compare the proportions of male and female students who smoke using a 95% confidence interval for the difference between the two proportions. Here's how to approach it: ### Step-by-Step Solution 1. **Extract Data from the Table:** | Gender | Smoke: Yes | Smoke: No | Total | |--------|-----------------------|-------| | Male | 9 | 51 | 60 | | Female | 9 | 81 90 | | Total | 18 | 132 | 150 | 2. **Calculate Proportions:** - Proportion of male smokers (\(p_1\)): \[ p_1 = \frac{9}{60} = .15 \] - Proportion of female smokers (\(p_2\)): \[ p_2 = \frac{9}{90} = .10 \] 3. **Calculate the Standard Error (SE) for the Difference in Proportions:** \[ SE = \sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2}} \] Where \(n_1 = 60\) and \(n_2 = 90\). \[ SE = \sqrt{\frac{.15 \times .85}{60} + \frac{.10 \times .90}{90}} \] \[ SE \approx \sqrt{\frac{.1275}{60} + \frac{.09}{90}} \] \[ SE \approx \sqrt{.002125 + .001} \] \[ SE \approx \sqrt{.003125} \] \[ SE \approx .0559 \] 4. **Calculate the 95% Confidence Interval:** - The critical value for a 95% confidence interval is approximately 1.96. - Calculate the margin of error (ME): \[ ME = 1.96 \times SE = 1.96 \times .0559 \approx .1095 \] - The confidence interval for \(p_1 - p_2\) is: \[ (p_1 - p_2) \pm ME = (.15 - .10) \pm .1095 \] \[ .05 \pm .1095 = (-.0595, .1595) \] ### Interpretation The 95% confidence interval for the difference in proportions of male and female students who smoke is approximately \((-.0595, .1595)\). This interval includes zero, suggesting there is no statistically significant difference between the proportions of male and female smokers at the 95% confidence level.