Final answer:
The confidence interval for an unknown population mean at a 95% confidence level will be longer for a smaller sample size (n=15) when compared to a larger sample size (n=24), because a smaller sample size results in a larger standard error.
Step-by-step explanation:
The question is asking to compare the length of 95% confidence intervals for a population mean (μ) when the sample size (n) is either 24 or 15, under the assumption that the population standard deviation (σ) is not known. Since a larger sample size tends to result in a narrower confidence interval for the same confidence level, the confidence interval would be longer (or wider) when the sample size is smaller (n=15) than when it is larger (n=24).
The length of the confidence interval depends on the standard error of the mean, which is calculated as the sample standard deviation (s) divided by the square root of the sample size (n), and multiplied by the z-score corresponding to the desired level of confidence. Since the population standard deviation is unknown, the t-distribution would be used rather than the normal distribution, and the t-score would replace the z-score. However, as the t-score is not provided in the question's context, we focus on the impact of sample size itself, where a smaller sample size will result in a larger standard error and thus a longer interval, assuming the same confidence level and that the standard deviation does not change significantly.