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 0.01 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

The following information is provided,
Significance Level, α = 0.05, Margin of Error, E = 0.01

Assume, The provided estimate of proportion p is, p = 0.5
The critical value for significance level, α = 0.05 is 1.96.

The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.5*(1 - 0.5)*(1.96/0.01)^2
n = 9604

b)
The provided estimate of proportion p is, p = 0.6
The critical value for significance level, α = 0.05 is 1.96.

The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.6*(1 - 0.6)*(1.96/0.01)^2
n = 9219.84

n = 9220

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