Question

Suppose that 307 out of a random sample of 375 letters mailed in the United States were delivered the day after they were mailed. Based on this, compute a 95% confidence interval for the proportion of all letters mailed in the United States that were delivered the day after they were mailed. Then complete the table below. Carry your intermediate computations to at least three decimal places. Round your answers to two decimal places. What is the lower limit of the 95% confidence interval? What is the upper limit of the 95% confidence interval? Х 5 ?

Answer 1

Solution :

Given that,

n = 375

x = 307

Point estimate = sample proportion = $\hat p$ = x / n = 307 /375 = 0.8187

1 - $\hat p$ = 1 - 0.8187 = 0.1813

At 95% confidence level

$\alpha$ = 1-0.95% =1-0.95 =0.05

$\alpha$/2 =0.05/ 2= 0.025

Z$\alpha$/2 = Z_{0.025 = 1.960}

Z_$\alpha$/2 = 1.960

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

= 1.960 * ($\sqrt$(0.8187*(0.1813) /375 )

= 0.039

A 95% confidence interval for population proportion p is ,

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

0.8187 - 0.039 < p < 0.8187 + 0.039

0.780 < p < 0.858

( 0.780, 0.858 )

Lower limit =0.780

Upper limit = 0.858