When sigma is unknown and you form a confidence interval for the mean with small samples, which distribution is used?

When sigma is unknown and you form a confidence interval for the mean with small samples, which distribution is used?

• A. Standard normal distribution
• B. t distribution
• C. F distribution
• D. Chi-square distribution

Answer: (B)

With sigma unknown and a small sample, you use the t distribution because you estimate the spread from the data. Replace sigma with the sample standard deviation S and form the statistic (X-bar − mu) / (S/√n). This statistic follows a t distribution with n − 1 degrees of freedom, reflecting the extra uncertainty introduced by estimating sigma from the sample. The t distribution has heavier tails than the standard normal, which makes the resulting confidence interval wider to account for that extra variability. As the sample size grows, S becomes a more accurate estimate of sigma, and the t distribution converges to the standard normal, so the intervals from the two approaches become similar. The other distributions aren’t appropriate for this specific task: the standard normal would require known sigma (or an effectively large sample), while the F and chi-square distributions are used for variance-related problems (like comparing variances or testing variance components) rather than constructing a confidence interval for a mean.

With sigma unknown and a small sample, you use the t distribution because you estimate the spread from the data. Replace sigma with the sample standard deviation S and form the statistic (X-bar − mu) / (S/√n). This statistic follows a t distribution with n − 1 degrees of freedom, reflecting the extra uncertainty introduced by estimating sigma from the sample.

The t distribution has heavier tails than the standard normal, which makes the resulting confidence interval wider to account for that extra variability. As the sample size grows, S becomes a more accurate estimate of sigma, and the t distribution converges to the standard normal, so the intervals from the two approaches become similar.

The other distributions aren’t appropriate for this specific task: the standard normal would require known sigma (or an effectively large sample), while the F and chi-square distributions are used for variance-related problems (like comparing variances or testing variance components) rather than constructing a confidence interval for a mean.