Bachelor’s Degree in Four Years 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 0.1 margin of error, and use a confidence level of 95%.

a. Assume that nothing is known about the percentage to be estimated.

Verified step by step guidance

Step 1: Identify the formula for determining the required sample size for estimating a population proportion. The formula is: n = (Z^2 * p * (1 - p)) / E^2, where n is the sample size, Z is the z-score corresponding to the confidence level, p is the estimated proportion, and E is the margin of error.

Step 2: Determine the z-score for a 95% confidence level. For a 95% confidence level, the z-score (Z) is approximately 1.96. This value is derived from the standard normal distribution.

Step 3: Since nothing is known about the percentage to be estimated, assume the most conservative estimate for p, which is 0.5. This maximizes the product p * (1 - p), ensuring the sample size is large enough.

Step 4: Substitute the values into the formula. Use Z = 1.96, p = 0.5, and E = 0.1. The formula becomes: n = (1.96^2 * 0.5 * (1 - 0.5)) / 0.1^2.

Step 5: Simplify the expression to calculate the required sample size. Perform the operations in the numerator and denominator step by step: square the z-score, calculate p * (1 - p), and divide by the square of the margin of error. This will give you the final sample size.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sample Size Determination

Sample size determination is a statistical process used to calculate the number of observations or replicates needed to ensure that the results of a study are reliable and valid. It involves considering factors such as the desired margin of error, confidence level, and the variability of the population. In this case, the goal is to estimate the percentage of students earning a degree within a specified margin of error.

Margin of Error

The margin of error is a statistic that expresses the amount of random sampling error in a survey's results. It indicates the range within which the true population parameter is expected to fall, given a certain confidence level. A smaller margin of error requires a larger sample size, which increases the precision of the estimate—in this scenario, a margin of error of 0.1 means the estimate should be within 10% of the true percentage.

Confidence Level

The confidence level is the probability that the confidence interval calculated from the sample data will contain the true population parameter. Common confidence levels are 90%, 95%, and 99%. A 95% confidence level, as specified in this question, means that if the same sampling method were repeated multiple times, approximately 95% of the calculated intervals would capture the true percentage of full-time college students earning a degree in four years.

Recommended video:

06:33

Introduction to Confidence Intervals

Related Practice

Textbook Question

Bachelor’s Degree The president of Brown University wants to estimate the mean time (years) it takes students to earn a bachelor’s degree. How many students must be surveyed in order to be 95% confident that the estimate is within 0.2 year of the true population mean? Assume that the population standard deviation is sigma=1.3 years

Textbook Question

Voting Survey In a survey of 1002 people, 70% said that they voted in a recent presidential election (based on data from ICR Research Group). Voting records show that 61% of eligible voters actually did vote.

d. Are the survey results consistent with the actual voter turnout of 61%? Why or why not?

Textbook Question

Bachelor’s Degree in Four Years 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 0.1 margin of error, and use a confidence level of 95%.

b. Assume that prior studies have shown that about 40% of full-time students earn bachelor’s degrees in four years or less.

Textbook Question

Alcohol in Children’s Movies Listed below is a simple random sample of times (seconds) that animated children’s movies showed the use of alcohol (based on Data Set 20 “Alcohol and Tobacco in Movies” in Appendix B).

a. Are the requirements for constructing a 95% confidence interval estimate of the population mean satisfied? If so, construct that confidence interval.