Question

Required information Leakage from underground fuel tanks has been a source of water pollution. In a random sample of 91 gasoline stations, 12 were found to have at least one leaking underground tank.

Find a 95% confidence interval for the proportion of gasoline stations with at least one leaking underground tank. Round the answers to three decimal places.

The 95% confidence interval is (  ,  ).

How many stations must be sampled so that a 95% confidence interval specifies the proportion to within ±0.05? (Round up the final answer to the nearest integer.)

Answer 1

Solution :

Given that,

Point estimate = sample proportion = $\hat p$ = x / n = 12 / 91 = 0.132

1 - $\hat p$ = 1 - 0.132 = 0.868

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

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

= 1.96 ($\sqrt$((0.132 * 0.868) / 91 )

= 0.070

A 95% confidence interval for population proportion p is ,

$\hat p$ ± E

= 0.132  ± 0.070

= ( 0.062, 0.202 )

b) margin of error = E = 0.05

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

= (1.96 / 0.05)² * 0.132 * 0.868

= 176.06

sample size = n = 177