Question

A researcher examined a random sample of 150 homes in a small city and found that 24 had solar panels installed on their roofs. Use the sample to construct a 99% confidence interval for the proportion of all homes in the city that have solar panels installed on their roofs.

What is the sample proportion? Round to two decimal places.

Answer 1

Solution :

Given that,

n = 150

x = 24

Point estimate = sample proportion = $\hat p$ = x / n = 0.16

1 - $\hat p$ = 0.84

At 99% confidence level the z is ,

$\alpha$ = 1 - 99% = 1 - 0.99 = 0.01

$\alpha$ / 2 = 0.01 / 2 = 0.005

Z_$\alpha$/2 = Z_0.005 = 2.576

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

= 2.576 * ($\sqrt$((0.16 * 0.84) / 150)

= 0.077

A 99% confidence interval for population proportion p is ,

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

0.16 - 0.077 < p < 0.16 + 0.077

0.083 < p < 0.237

A 99% confidence interval for the proportion of all homes in the city that have solar panels installed on their roofs. ( 0.083 , 0.237 )