Which distribution should be used to construct a confidence interval for the mean when the sample size is small and the degrees of freedom are $df=5$?

• A. The t-distribution
• B. The F-distribution
• C. The standard normal distribution
• D. The chi-square distribution

Answer: (A)

Verified step by step guidance

1. Understand the context: When constructing a confidence interval for the mean, the choice of distribution depends on the sample size and whether the population standard deviation is known.
2. Recall that for large sample sizes (usually n > 30) and known population standard deviation, the standard normal distribution (Z-distribution) is used.
3. For small sample sizes (n ≤ 30) and unknown population standard deviation, the sample standard deviation is used as an estimate, which introduces extra variability.
4. In this case, the sampling distribution of the sample mean follows a t-distribution with degrees of freedom equal to n - 1 (here, df = 5). The t-distribution accounts for the additional uncertainty due to estimating the standard deviation from a small sample.
5. Therefore, when the sample size is small and degrees of freedom are 5, the appropriate distribution to use for constructing the confidence interval for the mean is the t-distribution.