Ages of College Students An admissions director wants to estimate the mean age of all students enrolled at a college. The estimate must be within 1.5 years of the population mean. Assume the population of ages is normally distributed.
a. Determine the minimum sample size required to construct a 90% confidence interval for the population mean. Assume the population standard deviation is 1.6 years.

Verified step by step guidance

1. Step 1: Recall the formula for determining the minimum sample size required to estimate a population mean: n = (Z * σ / E)^2, where n is the sample size, Z is the critical value corresponding to the confidence level, σ is the population standard deviation, and E is the margin of error.
2. Step 2: Identify the given values from the problem: the confidence level is 90%, so the critical value Z can be found using a Z-table or standard normal distribution (Z ≈ 1.645 for 90% confidence level). The population standard deviation (σ) is 1.6 years, and the margin of error (E) is 1.5 years.
3. Step 3: Substitute the given values into the formula: n = (1.645 * 1.6 / 1.5)^2. Simplify the numerator and denominator inside the parentheses first.
4. Step 4: Square the result of the division to calculate the minimum sample size. Ensure that the final value of n is rounded up to the nearest whole number, as sample size must be an integer.
5. Step 5: Interpret the result: The calculated sample size represents the minimum number of students that must be sampled to ensure the estimate of the mean age is within 1.5 years of the population mean with 90% confidence.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from a sample, that is likely to contain the population parameter (like the mean) with a specified level of confidence. For example, a 90% confidence interval means that if we were to take many samples and construct intervals, approximately 90% of those intervals would contain the true population mean.

Sample Size Determination

Sample size determination involves calculating the number of observations needed to achieve a desired level of precision in estimating a population parameter. In this case, the formula incorporates the desired margin of error, the population standard deviation, and the critical value from the normal distribution corresponding to the confidence level.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. Many statistical methods, including confidence intervals, assume that the underlying population is normally distributed, which allows for the application of certain statistical techniques and the use of z-scores for calculations.