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 99​%. Complete parts​ (a) through​ (c) below. a. Assume that nothing is known about the percentage to be estimated. nequals nothing ​(Round up to the nearest​ integer.) b. Assume prior studies have shown that about 40​% of​ full-time stud ents earn​ bachelor's degrees in four years or less. nequals nothing ​(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) only slightly increases the sample size. B. ​No, using the additional survey information from part​ (b) only slightly reduces the sample size. C. ​Yes, using the additional survey information from part​ (b) dramatically 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 = 0.05

At 99% confidence level

= 1 - 99%

= 1 - 0.99 =0.01

/2 = 0.005

Z_/2 = Z_0.005 = 2.576  

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

= (2.576 / 0.05)² * 0.5 * 0.5

= 663.57

sample size = n = 664

b) = 0.40

1 - = 1 - 0.40 = 0.60

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

= (2.576 / 0.05)² * 0.40 * 0.60

= 637.03

sample size = n = 638

c) correct option is = C