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FRM Part I · FRM Exam Part I · Common Univariate Random Variables

A sample of n = 10 independent observations is drawn from a normal population. The statistic (n-1)s²/σ² is computed using the sample variance s² and the population variance σ². Which distribution does this statistic follow, and with how many degrees of freedom?

The statistic follows a chi-squared distribution with 9 degrees of freedom. For normal data, (n-1)s²/σ² is chi-squared with n-1 degrees of freedom, and one degree is lost because the sample mean is estimated from the same 10 observations.

  1. AChi-squared with 10 degrees of freedom
  2. BChi-squared with 9 degrees of freedomCorrect
  3. CStudent's t with 9 degrees of freedom
  4. DF with 9 and 10 degrees of freedom

Explanation

For a normal population, (n-1)s²/σ² follows a chi-squared distribution with n-1 degrees of freedom. Here n-1 = 9. Using 10 degrees of freedom ignores the one degree of freedom lost by estimating the mean with the sample mean.

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