An admissions director wants to estimate the mean age of all students enrolled at a college. The estimate must be within one year of the population mean. Assume the population of ages is normally distributed. Determine the minimum required sample size to construct a 80% confidence interval for the population mean. Assume the population standard deviation is 1.2 years
To determine the minimum required sample size for constructing an 80% confidence interval for the population mean, we can use the sample size formula:
n = (Z * σ / E)^2
Where:
n = sample size
Z = Z-score (which corresponds to the desired confidence level)
σ = population standard deviation
E = maximum error (margin of error)
In this case, the confidence level is 80%, so the corresponding Z-score can be obtained from a standard normal distribution table. For an 80% confidence level, the Z-score is approximately 1.28.
The given population standard deviation is 1.2 years, and the maximum error is 1 year (since the estimate must be within one year of the population mean).
Plugging in the values, we have:
n = (1.28 * 1.2 / 1)^2
n = (1.536 / 1)^2
n = (1.536)^2
n ≈ 2.3607
Since we cannot have a fraction of a person in a sample, we need to round up the sample size to the nearest whole number.
Therefore, the minimum required sample size to construct an 80% confidence interval for the population mean is 3.