Question

In a study of government financial aid for college students, it becomes necessary to estimate the percentage of full-time college students who earn a bachelor's degree in four years or less. Find the sample size needed to estimate that percentage. Use a 0.02 margin of error and use a confidence level of 95%. Complete parts (a) through (c) below. a. Assume that nothing is known about the percentage to be estimated. n = (Round up to the nearest integer.) b. Assume prior studies have shown that about 45% of full-time students earn bachelor's degrees in four years or less. n = (Round up to the nearest integer.) c. Does the added knowledge in part (b) have much of an effect on the sample size?

Answer 1

Solution,

Given that,

a) $\hat p$ =  1 - $\hat p$ = 0.5

margin of error = E = 0.02

At 95% confidence level

$\alpha$ = 1 - 95%

$\alpha$ = 1 - 0.95 =0.05

$\alpha$/2 = 0.025

Z_$\alpha$/2 = Z_0.025 = 1.96   

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

= (1.96 / 0.02 )² * 0.5 * 0.5

= 2401

sample size = n = 2401

b) $\hat p$ = 0.45

1 - $\hat p$ = 1 - 0.45 = 0.55

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

= (1.96 / 0.02 )² * 0.45 * 0.55

= 2376.99

sample size = n = 2377

c) Having an estimate of the population proportion reduces the minimum sample size is needed .