 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2013/o4/e/solution

σ^{2} > 0

S / σ^{2} \sim χ_{n}^{2}

. Therefore the power function is

w (σ^{2}) = 1 - F_{n} (\frac{σ _{0}^{2}}{σ ^{2}} F_{n}^{- 1} (1 - α)) .

The argument of

F_{n}

decreases as

σ^{2}

increases. Since the chi-squared density is positive on the positive half-line, the power is continuous and strictly increasing, with

w (0^{+}) = 0, w (σ_{0}^{2}) = α, w (\infty) = 1.

These features determine the requested rough sketch. The curve passes through the size at the null value and approaches one for very large variance.

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