Question

The following is a 90% confidence interval for p:(0.26, 0.54). How large was the sample used to construct this interval?

Answer 1

Solution :

given that

Lower confidence interval = 0.26

Upper confidence interval = 0.54

Point estimate = $\hat p$ = (Lower confidence interval + Upper confidence interval ) / 2

=  $\hat p$ = (0.26+0.54) / 2

Point estimate = $\hat p$ =0.4

Margin of error = E = Upper confidence interval - $\hat p$

=0.54 - 0.4

Margin of error = E = 0.14

Solution :

Given that,

$\hat p$ = 0.4

1 - $\hat p$ = 1 - 0.4 = 0.6

margin of error = E = 0.14

At 90% confidence level z

$\alpha$ = 1 - 90%  

$\alpha$ = 1 - 0.90 =0.10

$\alpha$/2 = 0.05

Z$\alpha$/2 = Z_{0.05 = 1.645}

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

= (1.645 / 0.14)² * 0.4 * 0.6

=33