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 0.04 margin of error and use a confidence level of 90​%. 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 60​% of​ full-time students earn​ bachelor's degrees in four years or less.

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) does not change the sample size.

B.

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

C.

​No, using the additional survey information from part​ (b) only slightly reduces 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 = 0.04

At 90% confidence level

= 1 - 90%

= 1 - 0.90 =0.10

/2 = 0.05

Z/2 = Z0.05  = 1.645

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

= (1.645 / 0.04)2 * 0.5 * 0.5

= 422.81

sample size = n = 423

b) = 0.60

1 - = 1 - 0.60 = 0.40

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

= (1.645 / 0.04)2 * 0.60 * 0.40

= 405.90

sample size = n = 406

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