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IAI Actuarial Core Principles · Actuarial Statistics · Confidence intervals and prediction intervals

X1,...,Xn are independent N(mu, sigma^2) with both parameters unknown. Which of the following is a pivotal quantity for mu that can be used to construct an exact confidence interval for any n?

The pivotal quantity is (Xbar - mu)/(S/sqrt(n)). It follows a t distribution with n-1 degrees of freedom regardless of the values of mu and sigma, so it gives an exact interval for mu when sigma is unknown.

  1. A(Xbar - mu)/(sigma/sqrt(n))
  2. B(Xbar - mu)/(S/sqrt(n))Correct
  3. C(Xbar - mu)/(S/n)
  4. D(n-1)S^2/sigma^2
  5. Xbar/S

Explanation

With sigma unknown, replacing sigma by S gives (Xbar - mu)/(S/sqrt(n)), which has a t distribution with n-1 degrees of freedom whatever mu and sigma are. The first option involves unknown sigma. The chi-square quantity concerns sigma^2, not mu. Xbar/S is not free of mu.

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