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.n=B.)Assume prior studies have shown that about 55%of​ full-time students earn​ bachelor's degrees in four years or less.n=c. Does the added knowledge in part​ (b) have much of an effect on the sample​ size?

Answer:a.   n=2401 studentsb.   n=2377 studentsc.   B. No, using the additional survey information from part​ (b) only slightly reduces the sample size.Step-by-step explanation:a. The sample size for a sample proportion about the mean is calculated using the formula:[tex]n=(\frac{z_{\alpha/2}}{E})^2p(1-p)[/tex]Where p is the proportion and E is the margin of error.-If nothing is known about the proportion to be studied, we use p=0.5:[tex]n=(\frac{z_{\alpha/2}}{E})^2p(1-p)\\\\=(1.96/0.02)^20.5(1-0.5)\\\\=2401[/tex]Hence, the required sample size is 2401b. If the proportion to be estimated is given, we substitute it for p in the formula.-Given p=0.55, the required sample size can be calculated as:[tex]n=(\frac{z_{\alpha/2}}{E})^2p(1-p)\\\\=(1.96/0.02)^20.55(1-0.55)\\\\=2376.99\approx2377[/tex]Hence, the required sample size for a given proportion of 55% is approximately 2377 studentsc. The added information in b had a reducing effect on the sample size:[tex]\bigtriangleup n=n_a-n_b\\\\=2401-2377\\\\=24[/tex]-The sample size slightly reduces by 24 students.Hence, No, using the additional survey information from part​ (b) only slightly reduces the sample size.