Question

Problem 1. A politician is interested how voters in her district feel about gun control. She surveyed 175 voters and found that 96 support gun control. She wants to estimate the proportion of all voters in her district who support gun control with a 95% confidence interval. (a) (2 points) Find the point estimate. (b) (2 points) Verify that the conditions for inference are satisfied. (c) (2 points) Construct the confidence interval for estimating the population proportion. Interpret your results. (d) (2 points) Without changing confidence, find the sample size needed to reduce the margin of error to 5%.

Answer 1

Solution :

Given that,

n = 175

x = 96

Point estimate = sample proportion = $\hat p$ = x / n = 96 / 175 = 0.549

1 - $\hat p$ = 1 - 0.549 = 0.451

At 95% confidence level

$\alpha$ = 1 - 95%

$\alpha$ =1 - 0.95 =0.05

$\alpha$/2 = 0.025

Z$\alpha$/2 = Z0.025 = 1.960

Margin of error = E = Z$\alpha$ / 2 * $\sqrt$ (($\hat p$ * (1 - $\hat p$ )) / n)

= 1.96 ($\sqrt$(( 0.549 * 0.451) / 175 )

= 0.074

A 95% confidence interval for population proportion p is ,

$\hat p$ - E < p < $\hat p$ + E

0.549 - 0.074 < p < 0.549 + 0.074

0.475 < p < 0.623

( 0.475 , 0.623 )

The 95% confidence interval for the population proportion p is : - ( 0.475 , 0.623 )

Interpret : - About a 95% confidence interval the true estimate the proportion of all voters in her district who support gun control

( d )

n = 175

x = 96

$\hat p$ = 96 / 175 = 0.0.549

1 - $\hat p$ = 1 - 0.549 = 0.451

margin of error = E = 5% = 0.05

At 95% confidence level

$\alpha$ = 1 - 95%  

$\alpha$ = 1 - 0.95 =0.05

$\alpha$/2 = 0.025

Z$\alpha$/2 = Z0.025  = 1.96

sample size = n = (Z$\alpha$ / 2 / E )2 * $\hat p$ * (1 - $\hat p$ )

= (1.96 / 0.05 )2 * 0.549 * 0.451

= 380.47

Sample size = n = 381