When conducting interval estimation of μ from a normally distributed population with a sample of 30, which distribution should be used?

When conducting interval estimation for the population mean μ from a normally distributed population, specifically with sample sizes less than 30 or when the population standard deviation is unknown, the t-distribution is the appropriate distribution to use. In this case, since the sample size is 30, which is reasonably large, you might wonder whether to use the normal distribution. However, routine practice suggests using the t-distribution when the population standard deviation is not known, which is common in practical applications.

The t-distribution accounts for the additional variability that arises from estimating the population standard deviation from the sample. This is particularly important with smaller sample sizes, as the t-distribution has heavier tails, which provides a more accurate estimate of confidence intervals. Even with a sample size of 30, it is still conservative to use the t-distribution, especially since the exact population standard deviation is typically unknown.

In this scenario, utilizing the t distribution with 29 degrees of freedom (derived from n-1, where n is the sample size) is the correct choice for constructing the confidence interval for the population mean. The heavier tails of the t-distribution help ensure a more reliable range for estimating μ, ensuring that the interval is appropriate given the conditions of the problem.