Question

Suppose the U.S. President wants an estimate of the proportion of the population who support his current policy toward revisions in the Social Security system. The president wants the estimate to be within 0.02 of the true proportion. Assume a 90 percent level of confidence. The presidential polotical advisors found a similair survery from two years agothat reported 61% of people supoorted health care revisions. .

a. how large a sample is required? (Round intermediate values to 3 decimal points. Round up your answer to the next whole number)

b. how large a sample would be necessary if no estimate were available for the proportion supporting current policy? (Round intermediate values to 3 decimal points. Round up your answer to the next whole number)

Answer 1

Solution :

Given that,

Margin of error = E = 0.02

At 90% confidence level the z is ,

$\alpha$ = 1 - 90% = 1 - 0.90 = 0.10

$\alpha$ / 2 = 0.10 / 2 = 0.05

Z_$\alpha$/2 = Z_0.05 = 1.645

(a)

$\hat p$ = 61% = 0.61

1 - $\hat p$ = 1 - 0.61= 0.39

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

= (1.645 / 0.02)² * 0.61 * 0.39

= 1609.40

Sample size = n = 1610

(b)

$\hat p$ = 61% = 0.5

1 - $\hat p$ = 1 - 0.5 = 0.5

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

= (1.645 / 0.02)² * 0.5 * 0.5

= 1691.26

Sample size = n = 1692