7. Sampling Distributions & Confidence Intervals: Mean

Introduction to Confidence Intervals

7. Sampling Distributions & Confidence Intervals: Mean

Introduction to Confidence Intervals

2:24 minutes

Problem 7.r.1a

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%.

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.

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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.

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Textbook Question

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