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 90​%. Complete parts​ (a) through​ (c) below.

a. Assume that nothing is known about the percentage to be estimated. n=______d 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?

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) dramatically reduces the sample size.

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

Answer 1

Part a

The sample size formula is given as below:

n = p*q*(Z/E)^2

We are not given the estimate for proportion, so we take

p = 0.5

q = 1 – p = 0.5

Confidence level = 90%

Critical Z value = 1.6449

(by using z-table)

Margin of error = E = 0.02

The sample size is given as below:

n = p*q*(Z/E)^2

n = 0.5*0.5*(1.6449/0.02)^2

n = 1691.06

Required sample size = 1692

Part b

The sample size formula is given as below:

n = p*q*(Z/E)^2

We are not given the estimate for proportion, so we take

p = 0.45

q = 1 – p = 0.55

Confidence level = 90%

Critical Z value = 1.6449

(by using z-table)

Margin of error = E = 0.02

The sample size is given as below:

n = p*q*(Z/E)^2

n = 0.45*0.55*(1.6449/0.02)^2

n = 1674.149

Required sample size = 1675

Part c

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