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FRM Part I · FRM Exam Part I · Hypothesis Testing

An analyst builds a 95% confidence interval for a mean using a sample of 25 observations from a normally distributed population with unknown variance. Relative to an interval built using the standard normal critical value of 1.96 with the same sample standard deviation, the correct interval based on the t-distribution with 24 degrees of freedom will be:

The t-based interval is wider. When the variance is estimated from a small sample, the t-distribution has fatter tails than the normal, so its 95% critical value (about 2.06 for 24 degrees of freedom) exceeds 1.96, widening the interval.

  1. Anarrower, because the t-distribution has less probability in its tails
  2. Bthe same width, because the sample mean is unbiased
  3. Cwider, because the t-distribution has fatter tails and the critical value exceeds 1.96Correct
  4. Dnarrower, because the sample standard deviation is estimated from data

Explanation

With unknown variance and a small sample, the t statistic is used. The t critical value with 24 degrees of freedom is about 2.06, larger than 1.96, reflecting extra uncertainty from estimating the standard deviation. The interval is therefore wider.

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