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

Verified step by step guidance

Step 1: Identify the key components of the problem. The confidence level is 95%, the margin of error (E) is 0.2 years, the population standard deviation (σ) is 1.3 years, and we need to determine the required sample size (n).

Step 2: Recall the formula for determining the sample size for estimating a population mean: n = (Z * σ / E)^2. Here, Z is the critical value corresponding to the desired confidence level, σ is the population standard deviation, and E is the margin of error.

Step 3: Determine the critical value (Z) for a 95% confidence level. For a 95% confidence level, the Z-value is approximately 1.96. This value is obtained from the standard normal distribution table.

Step 4: Substitute the known values into the formula. Replace Z with 1.96, σ with 1.3, and E with 0.2 in the formula: n = (1.96 * 1.3 / 0.2)^2.

Step 5: Simplify the expression inside the parentheses and then square the result to calculate the required sample size. Ensure the final value of n is rounded 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 needed to achieve a desired level of confidence and precision in estimating a population parameter. In this case, it involves using the formula that incorporates the desired margin of error, the population standard deviation, and the critical value from the normal distribution corresponding to the confidence level.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence. For example, a 95% confidence interval suggests that if we were to take many samples and build intervals, approximately 95% of those intervals would contain the true mean. This concept is crucial for understanding how precise our estimate is.

Population Standard Deviation

Population standard deviation is a measure of the amount of variation or dispersion in a set of values. It quantifies how much individual data points differ from the population mean. In this scenario, knowing the population standard deviation (sigma = 1.3 years) is essential for calculating the sample size needed to ensure that the estimate of the mean time to earn a bachelor’s degree is accurate within the specified margin of error.

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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

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

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.