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 0.05 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.

___ ​(Round up to the nearest​ integer.)

b. Assume prior studies have shown that about 60​% of​ full-time students earn​ bachelor's degrees in four years or less.

___ (Round up to the nearest​ integer.)

c. Does the added knowledge in part​ (b) have much of an effect on the sample​ size?

A. No, using the additional survey information from part​ (b) only slightly reduces the sample size.

B. No, using the additional survey information from part​ (b) does not change the sample size.

C. Yes, using the additional survey information from part​ (b) only slightly increases the sample size.

D. Yes, using the additional survey information from part​ (b) dramatically reduces the sample size.

Answer 1

Solution,

Given that,

a) =  1 - = 0.5

margin of error = E = 0.05

At 95% confidence level

= 1 - 95%  

= 1 - 0.95 =0.05

/2 = 0.025

Z/2 = Z0.025  = 1.96

sample size = n = (Z / 2 / E )2 * * (1 - )

= (1.96 /0.05 )2 * 0.5 * 0.5

= 384.16

sample size = n = 385

b) = 0.60

1 - = 1 - 0.60 = 0.40

sample size = n = (Z / 2 / E )2 * * (1 - )

= (1.96 /0.05 )2 * 0.60 * 0.40

= 368.79

sample size = n = 369

c) D) Yes, using the additional survey information from part (b) dramatically reduces the sample size.