Normal Distribution

Instead of the rather odd Example 8.1.3 in DeGroot and Schervish, we calculate power functions for the test about the mean of a normal distribution with known variance (Example 8.1.2 in DeGroot and Schervish).

In both sections below we use θ for the true (unknown) parameter value (the mean of the distribution of the data) and μ0 for the parameter value hypothesized under the null hypothesis.

One-Sided Alternative

Suppose we choose

T = (Xn &minus μ0) ⁄ (&sigma ⁄ √n)

as our test statistic (so we are doing an upper-tailed test). Suppose we reject H0: θ = &mu0 when T ≥ 1.645

The power function is then

π(θ) = Prθ(T ≥ 1.645)

The distribution of T is normal with mean

Eθ(T) = (θ &minus μ0) ⁄ (&sigma ⁄ √n)

and variance

varθ(T) = σ2

Thus the power function is calculated and graphed as follows.

Changing the values of σ and n will change the curve.

Two-Sided Alternative

Suppose we now choose
T = |(Xn &minus μ0) ⁄ (&sigma ⁄ √n)|

as our test statistic (the absolute value of the test statistic from the preceding section).

Now reject when T ≥ c is the same as rejecting when

Told ≤ − c or Told ≥ c

where Told is the test statistic of the preceding section. So the power involves the probability for two tails

As before, changing the values of σ and n will change the curve.

Student T Distribution

Here we look at the power function for t tests, following pp. 488–490 in DeGroot and Schervish.

Now the test statistic is

T = (Xn &minus μ0) ⁄ (sn ⁄ √n)

(so we are doing an upper-tailed test).

When the null hypothesis is false, the test statistic has a noncentral t distribution. If we write

T = (Xn &minus μ0) ⁄ (σ ⁄ √n) × (σ ⁄ sn)
Then the normal part (the first term) has mean
ψ = (θ &minus μ0) ⁄ (σ ⁄ √n)
when the true mean is θ and the true variance is σ2. The other part (the second term) is one over the square root of a chi-square on n − 1 degrees of freedom. Thus the distribution of the test statistic is noncentral chi-square (DeGroot and Schervish, p. 488) with noncentrality parameter ψ and n − 1 degrees of freedom.

One-Sided Alternative

Thus the power function for an upper-tailed test is calculated and graphed as follows.

I have no idea what these warnings are about. Apparently, the calculation of the noncentral t probabilities is not as accurate as it could be, although presumably accurate enough for drawing this curve.

Changing the values of σ and n will change the curve.

Two-Sided Alternative

And the power function for a two-tailed test is calculated and graphed as follows.

I have no idea what these warnings are about. Apparently, the calculation of the noncentral t probabilities is not as accurate as it could be, although presumably accurate enough for drawing this curve.

Changing the values of σ and n will change the curve.