Final answer:
To estimate the percentage of full-time college students who earn a bachelor's degree in four years or less, we need to determine the sample size needed with a 0.04 margin of error and a confidence level of 99%. When prior studies have shown that about 60% of full-time students earn bachelor's degrees in four years or less, the sample size needed is 410 (rounded up to the nearest integer). The added knowledge in part (b) only slightly reduces the sample size.
Step-by-step explanation:
To estimate the percentage of full-time college students who earn a bachelor's degree in four years or less, we need to determine the sample size needed with a 0.04 margin of error and a confidence level of 99%.
a) When nothing is known about the percentage, the sample size needed can be calculated using the formula:
n = (Z^2 * p * (1-p)) / (E^2)
where Z is the z-score corresponding to the desired confidence level (2.576 for 99%), p is the estimated proportion, and E is the margin of error.
Using the formula, when nothing is known about the percentage, the sample size needed is 664 (rounded up to the nearest integer).
b) When prior studies have shown that about 60% of full-time students earn bachelor's degrees in four years or less, the sample size needed can be calculated using the same formula, but with the estimated proportion of 60%.
Using the formula, when prior studies have shown that about 60% of full-time students earn bachelor's degrees in four years or less, the sample size needed is 410 (rounded up to the nearest integer).
c) The added knowledge in part (b) does have an effect on the sample size, but it only slightly reduces the sample size. The decrease in sample size is due to having a more accurate estimate of the proportion (60%) compared to having no prior knowledge (where the estimate could range from 0% to 100%).
Therefore, the correct answer is (b) No, using the additional survey information from part (b) only slightly reduces the sample size.