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 0.03 margin of error and use a confidence level of 99​%.

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.

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.

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

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

Answer 1

Solution :

Given that,

a) = 1 - = 0.5

margin of error = E = 0.03

At 99% confidence level

= 1 - 99%

=1 - 0.99 =0.01

/2 = 0.005

Z/2 = 2.576

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

= (2.576 / 0.03)² * 0.5 * 0.5

= 1843.27

sample size = n = 1844

b) = 0.60

1 - = 1 - 0.60 = 0.40

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

= (2.576 / 0.03)² * 0.60 * 0.40

= 1769.54

sample size = n = 1770

c) correct option is = A.

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