7. Sampling Distributions & Confidence Intervals: Mean

Introduction to Confidence Intervals

7. Sampling Distributions & Confidence Intervals: Mean

Introduction to Confidence Intervals

2:28 minutes

Problem 7.r.1b

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.

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 population proportion, and E is the margin of error.

Step 2: Determine the values for the variables in the formula. From the problem, the confidence level is 95%, so the z-score (Z) corresponding to this confidence level is approximately 1.96. The estimated population proportion (p) is 0.40, and the margin of error (E) is 0.1.

Step 3: Substitute the values into the formula. Replace Z with 1.96, p with 0.40, and E with 0.1 in the formula: n = (1.96^2 * 0.40 * (1 - 0.40)) / 0.1^2.

Step 4: Simplify the numerator of the formula. Calculate Z^2 (1.96^2), p * (1 - p) (0.40 * 0.60), and multiply these values together.

Step 5: Divide the result from Step 4 by the square of the margin of error (E^2 = 0.1^2). This will give you the required sample size (n). Round up to the nearest whole number, as sample size must be an integer.

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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 a study's results are reliable and valid. It involves considering factors such as the desired margin of error, confidence level, and the estimated proportion of the population. In this case, the goal is to estimate the percentage of students earning degrees within a specific timeframe, which requires an appropriate sample size to achieve accurate results.

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, as it reflects a higher precision in estimating the population proportion—in this scenario, the percentage of students graduating in four years.

Confidence Level

The confidence level is the probability that the value of a parameter falls within a specified range of values. Commonly expressed as a percentage, such as 95%, it indicates the degree of certainty researchers have in their estimates. A 95% confidence level means that if the same study were repeated multiple times, 95% of the calculated confidence intervals would contain the true population parameter, making it a standard choice in statistical analysis.

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