Question

A simple random sample of 400 individuals provides 300 Yes responses. a. What is the point estimate of the proportion of the population that would provide Yes responses (to 2 decimals)? Later use p rounded to 2 decimal places. b. What is your estimate of the standard error of the proportion (to 4 decimals)? c. Compute the 95% confidence interval for the population proportion (to 4 decimals)

Answer 1

Solution :

Given that,

n = 400

x = 300

a. Point estimate = sample proportion = $\hat p$ = x / n = 300 / 400 = 0.75

1 - $\hat p$ = 1 - 0.75 = 0.25

At 95% confidence level the z is ,

$\alpha$ = 1 - 95% = 1 - 0.95 = 0.05

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

Z_$\alpha$/2 = Z_0.025 = 1.96

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

= 1.96 * ($\sqrt$((0.75 * 0.25) / 400)

= 0.0424

c. A 95% confidence interval for population proportion p is ,

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

0.75 - 0.0424 < p < 0.75 + 0.0424

0.7076 < p < 0.7924

The 95% confidence interval for the population proportion p is : ( 0.7076 , 0.7924 )