Construct the indicated confidence interval for the difference between population proportions p1 - p2. Assume that the samples are independent and that they have been randomly selected.
-$\mathrm { x } _ { 1 } = 27 , \mathrm { n } _ { 1 } = 67$ and $\mathrm { x } _ { 2 } = 27 , \mathrm { n } _ { 2 } = 72 ;$ Construct a $95 \%$ confidence interval for the difference between population proportions $\mathrm { P } 1 - \mathrm { P } 2$.
A) $0.210

Question

Construct the indicated confidence interval for the difference between population proportions p1 - p2. Assume that the samples are independent and that they have been randomly selected.
-$\mathrm { x } _ { 1 } = 27 , \mathrm { n } _ { 1 } = 67$ and $\mathrm { x } _ { 2 } = 27 , \mathrm { n } _ { 2 } = 72 ;$ Construct a $95 \%$ confidence interval for the difference between population proportions $\mathrm { P } 1 - \mathrm { P } 2$.
A) $0.210 < \mathrm { p } 1 - \mathrm { p } 2 < 0.596$
B) $- 0.165 < \mathrm { p } 1 - \mathrm { p } 2 < 0.596$
C) $- 0.134 < \mathrm { p } _ { 1 } - \mathrm { p } _ { 2 } < 0.190$
D) $0.241 < \mathrm { p } 1 - \mathrm { p } 2 < 0.565$

Quizplus · Accepted Answer

The confidence interval for the difference between two population proportions is calculated using the formula: $$CI = (\hat{p}_1 - \hat{p}_2) \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$$ where $\hat{p}_1 = \frac{x_1}{n_1}$, $\hat{p}_2 = \frac{x_2}{n_2}$, and $Z_{\alpha/2}$ is the Z-score for the desired confidence level (for 95%, $Z_{\alpha/2} = 1.96$). Plugging in the given values, we find the confidence interval for the difference in proportions, which matches option C.

Construct the Indicated Confidence Interval for the Difference Between Population x1=27,n1=67\mathrm { x } _ { 1 } = 27 , \mathrm { n } _ { 1 } = 67x1​=27,n1​=67

Construct the Indicated Confidence Interval for the Difference Between Population $\mathrm { x } _ { 1 } = 27 , \mathrm { n } _ { 1 } = 67$