Question

The ways that we communicate have been dramatically affected by use of answering machines, fax machines, voice mail, and email. Suppose a sociologist wants to determine the current percentage of U.S. households using email. How many households must be surveyed in order to be 90% confident that the sample percentage is in error by no more than 4%?

a.) Assume that there is no available information that can be used. Find the conservative estimate of the sample size.

b.) Assume that in 1997, 17% of U.S. households used email.

Answer 1

Solution,

Given that,

a) $\hat p$ =  1 - $\hat p$ = 0.5

margin of error = E = 0.04

At 90% confidence level

$\alpha$ = 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 / 0.04)² * 0.5 * 0.5

= 422.81

sample size = n = 423

b) $\hat p$ = 0.17

1 - $\hat p$ = 1 - 0.17 = 0.83

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

= (1.645 / 0.04)² * 0.17 * 0.83

= 238.63

sample size = n = 239