Question

Construct a 95% confidence interval of the population proportion using the given information. x = 175, n = 250 The lower bound is The upper bound is . (Round to three decimal places as needed.)

Answer 1

Solution :

Given that,

n = 250

x = 175

Point estimate = sample proportion = $\hat p$ = x / n = 175/250=0.7

1 - $\hat p$ = 1 -0.7=0.3

At 95% confidence level

$\alpha$ = 1 - 95%  

$\alpha$ = 1 - 0.95 =0.05

$\alpha$/2 = 0.025

Z$\alpha$/2 = Z_{0.025 = 1.96}

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

= 1.96 ($\sqrt$((0.7*0.3) / 250)

= 0.057

A 95% confidence interval for population proportion p is ,

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

0.7- 0.057< p <0.7 + 0.057

0.643< p < 0.757

The 95% confidence interval for the population proportion p is : lower limit =0.643 , upper limit =0.757