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 95​%.

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 45​% of​ full-time students 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) does not change 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) only slightly reduces the sample size.

Answer 1

a) =  1 - = 0.5

margin of error = E = 0.03

At 95% confidence level

= 1 - 95%

= 1 - 0.95 =0.05

/2 = 0.025

Z_/2 = Z_0.025 = 1.96   

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

= (1.96 / 0.03)² * 0.5 * 0.5

= 1067.11

sample size = n = 1068

b) = 0.45

1 - = 1 - 0.45 = 0.55

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

= (1.96 / 0.03)² * 0.45 * 0.55

= 1056.44

sample size = n = 1057

c) correct option is = C