Construct a 95% confidence interval for $\mathrm { p } _ { 1 } - \mathrm { p } _ { 2 }$ for a survey that finds 30% of 240 randomly selected males and 41% of 200 randomly selected females are opposed to the death penalty.
A)(-0.200, -0.021)
B)(-1.532, 1.342)
C)(-0.561, 0.651)
D)(-1.324, 1.512)

Question

Quizplus · Accepted Answer

To construct a 95% confidence interval for the difference between two proportions, $\mathrm { p } _ { 1 } - \mathrm { p } _ { 2 }$, we use the formula:$$\left( \hat { p } _ { 1 } - \hat { p } _ { 2 } \right) \pm z * \sqrt { \frac { \hat { p } _ { 1 } ( 1 - \hat { p } _ { 1 } ) } { n _ { 1 } } + \frac { \hat { p } _ { 2 } ( 1 - \hat { p } _ { 2 } ) } { n _ { 2 } } }$$where $\hat { p } _ { 1 }$ and $\hat { p } _ { 2 }$ are the sample proportions, $n _ { 1 }$ and $n _ { 2 }$ are the sample sizes, and $z$ is the z-score corresponding to the desired confidence level (for 95% confidence, $z \approx 1.96$).Given:- $\hat { p } _ { 1 } = 0.30$ (30% of males)- $n _ { 1 } = 240$- $\hat { p } _ { 2 } = 0.41$ (41% of females)- $n _ { 2 } = 200$Plugging these values into the formula:$$\left( 0.30 - 0.41 \right) \pm 1.96 * \sqrt { \frac { 0.30 * 0.70 } { 240 } + \frac { 0.41 * 0.59 } { 200 } }$$$$= -0.11 \pm 1.96 * \sqrt { 0.00072917 + 0.0011965 }$$$$= -0.11 \pm 1.96 * \sqrt { 0.00192567 }$$$$= -0.11 \pm 1.96 * 0.04388$$$$= -0.11 \pm 0.086$$$$= (-0.196, -0.024)$$Rounding to three decimal places, the closest match is choice A) (-0.200, -0.021).

Construct a 95% Confidence Interval For p1−p2\mathrm { p } _ { 1 } - \mathrm { p } _ { 2 }p1​−p2​

Construct a 95% Confidence Interval For $\mathrm { p } _ { 1 } - \mathrm { p } _ { 2 }$