Write the word or phrase that best completes each statement or answers the question.
Use the traditional method to test the given hypothesis. Assume that the samples are independent and that they have been randomly selected
-In a random sample of 300 women, $45 \%$ favored stricter gun control legislation. In a random sample of 200 men, $25 \%$ favored stricter gun control legislation. Construct a $98 \%$ confidence interval for the difference between the population proportions $\mathrm { p } 1 - \mathrm { P } 2$.
A) $0.092

Question

Write the word or phrase that best completes each statement or answers the question.
Use the traditional method to test the given hypothesis. Assume that the samples are independent and that they have been randomly selected
-In a random sample of 300 women, $45 \%$ favored stricter gun control legislation. In a random sample of 200 men, $25 \%$ favored stricter gun control legislation. Construct a $98 \%$ confidence interval for the difference between the population proportions $\mathrm { p } 1 - \mathrm { P } 2$.
A) $0.092 < \mathrm { p } 1 - \mathrm { p } 2 < 0.308$
B) $0.114 < \mathrm { p } _ { 1 } - \mathrm { p } 2 < 0.286$
C) $0.102 < \mathrm { p } 1 - \mathrm { p } 2 < 0.298$
D) $0.118 < \mathrm { p } 1 - \mathrm { p } 2 < 0.282$

Quizplus · Accepted Answer

We will use the formula for the difference between two proportions in two independent samples:
$$\mathrm { p } _ { 1 } - \mathrm { P } _ { 2 } \pm z ^ { * } \sqrt { \frac { \mathrm { p } _ { 1 } ( 1 - \mathrm { p } _ { 1 } ) } { n _ { 1 } } + \frac { \mathrm { p } _ { 2 } ( 1 - \mathrm { p } _ { 2 } ) } { n _ { 2 } } }$$

where:
$\mathrm { p } _ { 1 }$ = proportion of women in favor of stricter gun control legislation 
$\mathrm { P } _ { 2 }$= proportion of men in favor of stricter gun control legislation 
$n _ { 1 }$ = sample size of women 
$n _ { 2 }$ = sample size of men 
$z ^ { * }$ = the z-score corresponding to a 98% confidence level, which is 2.33

Plugging in the values, we get:
$$\mathrm { p } _ { 1 } - \mathrm { P } _ { 2 } \pm 2.33\sqrt { \frac { 0.45 ( 1 - 0.45 ) } { 300 } + \frac { 0.25 ( 1 - 0.25 ) } { 200 } }$$
$$\mathrm { p } _ { 1 } - \mathrm { P } _ { 2 } \pm 0.102$$

Calculating the upper and lower bounds:
$$\mathrm { p } _ { 1 } - \mathrm { P } _ { 2 } + 0.102=0.298$$
$$\mathrm { p } _ { 1 } - \mathrm { P } _ { 2 } - 0.102=0.102$$

Therefore, the 98% confidence interval for the difference between the population proportions is (0.102, 0.298). Choice C is the best answer.

Write the Word or Phrase That Best Completes Each Statement 45%45 \%45%

Write the Word or Phrase That Best Completes Each Statement $45 \%$