Question

assuming data come from a random sample, under which of the following conditions should we not calculate a confidence interval for a population mean? a. population is normally distributed and sample size is 50 individuals. b. population is normally distributed and sample size is 20 individuals. c. population distribution is unknown and sample size is 50 individuals. d. population distribution is unknown and sample size is 20 individuals.

Explanation:

When the population distribution is unknown, we rely on the Central - Limit Theorem. For a small sample size (usually $n < 30$), if the population distribution is unknown, we cannot assume approximate normality of the sampling distribution of the sample mean. So, we should not calculate a confidence interval for the population mean. In options A and B, the population is normally distributed, so confidence intervals can be calculated. In option C, although the population distribution is unknown, the sample size $n = 50\geq30$, and by the Central - Limit Theorem, the sampling distribution of the sample mean is approximately normal. In option D, with an unknown population distribution and a small sample size ($n = 20<30$), we cannot calculate a reliable confidence interval for the population mean.

Answer:

D. Population distribution is unknown and sample size is 20 individuals.