Mathematics/Stats

Question

In a study of government financial aid for college students, it becomes necessary to estimate the percentage of -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%.

A.) Assume that nothing is known about the

percentage to be estimated.

B.)Assume prior studies have shown that

about 55% of full-time students earn bachelor's

degrees in four years or less.

c. Does the added knowledge in part (b)

have much of an effect on the sample size?

Answer 1

The equation for determining sample size n when you aren't given the point estimate (p-hat) is this:

n = za/2^2 *0.25 / E^2

You must first find the critical value za/2. The confidence level is 95% so the alpha a is 0.05, because together they must always equal 100%.

This is a two tailed test, so divide alpha in half (0.025), then subtract it from 1 (1 - 0.025 = 0.975 . Look up the resulting value in the Positive (Right Tail) chart to find the corresponding z-score. The z-score is 1.96.

Now square the z-score. 1.96^2 is 3.8416. Multiply this number by 0.25. This gives you 0.9604.

The given margin of error E is 0.01. Square it (0.0001). Now divide 0.09604 by 0.0001. This gives you 9,603.647. Round up because you can't have a partial person. This is your sample size n.

a. So you need 9,604 students to estimate that percentage.

For b. You have been given 55%. This will be your point estimate p-hat. So convert it to decimal.

The equation for estimating sample size when you have p-hat is this. n = za/2^2(p-hat * q-hat) / E^2

We know the za/2^2 is 3.8416. p-hat is 0.55. To find q-hat is the complement of p-hat so to calculate it, you subtract p-hat from 1. 1 - 0.55 is 0.45. So multiply 0.55 * 045 to get 0.2475. Multiply 3.8416 * 0.2475. This is 0.950796. Now divide that by your E^2 of 0.0001 and you get 9507.611. Round up.

b. You need 9508 people to estimate that percentage.

c. In this case, I would say it doesn't have much effect because you need a similar sample size.