Deriving Power for a One-Sided Z Test

From a Rejection Rule to Power

A one-sided z test has a clear rejection rule: reject when the test statistic falls far enough into the tail specified by the alternative hypothesis. Power asks a different question from the test itself: if the population mean actually has a particular alternative value, how often would this rule reject the null hypothesis? The answer follows by applying the normal sampling distribution under that alternative to the test’s rejection boundary.

The calculation depends on the test’s direction, significance level, sample size, known population standard deviation, and the alternative mean being considered. Power is therefore not a single property of a test independent of the actual effect. It is evaluated at a specified alternative.

Model and Critical Values

Consider a one-sample test about a population mean \(\mu\), with null boundary \(\mu_0\). Let \(X_1,\ldots,X_n\) be an independent random sample, and suppose the population standard deviation \(\sigma\) is known. The population is normally distributed, so the sample mean has an exact normal sampling distribution. If the population is not normal, a normal approximation for the sample mean may be appropriate for a sufficiently large sample, but that is an approximation rather than the exact model.

Under a population mean \(\mu\), the sample mean has standard error \(\sigma/\sqrt{n}\) and sampling distribution \(N(\mu,\sigma^2/n)\). The test statistic is \(Z=(\bar{x}-\mu_0)/(\sigma/\sqrt{n})\). Write \(z_{1-\alpha}\) for the standard normal value with probability \(1-\alpha\) to its left; equivalently, if \(Z_0\sim N(0,1)\), then \(P(Z_0\leq z_{1-\alpha})=1-\alpha\).

The hypotheses determine which tail defines rejection. A power calculation must use that same tail; changing the sign or using the other tail would calculate the probability of a different decision rule.

Deriving Power for an Upper-Tailed Test

For an upper-tailed test, keep the rejection boundary fixed and evaluate the chance of crossing it when the true mean is \(\mu_1>\mu_0\). Under that alternative, the sample mean is centered at \(\mu_1\), not at \(\mu_0\). Standardizing the boundary using the alternative sampling distribution gives the power formula.

For a complete derivation, let \(SE=\sigma/\sqrt{n}\). Under the alternative \(\mu=\mu_1\), the variable \((\bar{x}-\mu_1)/SE\) has the standard normal distribution. Therefore:

Here \(Z_0\sim N(0,1)\), and \(\Phi(a)=P(Z_0\leq a)\) is the standard normal cumulative distribution function. The last equality follows from \(P(Z_0>a)=1-\Phi(a)=\Phi(-a)\). This derivation shows why power increases when the specified alternative mean moves farther above the null boundary: the rejection boundary becomes less extreme relative to the alternative sampling distribution.

Power for a Lower-Tailed Test

The reasoning is parallel for a lower-tailed test, but its rejection region is on the low side of the null boundary. At a specified \(\mu_1<\mu_0\), the rejection event is \(\bar{x}<\mu_0-z_{1-\alpha}SE\). Standardizing that boundary under the distribution centered at \(\mu_1\) gives the following result.

To see the standardization directly, the power is \(P_{\mu_1}(\bar{x}<\mu_0-z_{1-\alpha}SE)\). Subtracting \(\mu_1\) and dividing by \(SE\) makes the cutoff \((\mu_0-\mu_1)/SE-z_{1-\alpha}\), so the probability is \(\Phi\) of that cutoff.

How the Distance to the Boundary Matters

Power is governed by how far the specified alternative lies from the null boundary in standard-error units, relative to the critical value. For an upper-tailed test, the relevant distance is \((\mu_1-\mu_0)/SE\); for a lower-tailed test, it is \((\mu_0-\mu_1)/SE\). If that distance is close to \(z_{1-\alpha}\), the alternative distribution is centered near the rejection boundary and power can be near one half. If it is much larger, most of the alternative sampling distribution falls in the rejection region.

Common Mistake and Interpretation

A frequent mistake is to calculate power by using the null distribution for both the rejection boundary and the probability. The null distribution determines the critical value, because that is how the significance level is set. But power is evaluated under the specified alternative, so the sampling distribution must be centered at \(\mu_1\). For example, in the first calculation, \(104.11225\) mL is the same fixed rejection boundary under either mean, but the chance of exceeding it is calculated from the distribution centered at \(108\) mL.

Another error is to report power without naming the alternative. A statement such as “the test has 94% power” is incomplete unless the conditions and the value of \(\mu_1\) are clear. A suitable interpretation says that, under the stated model and at the specified alternative mean, the test would reject \(H_0\) in about the stated proportion of repeated samples of the same size.

Power is useful when assessing how likely a test is to detect an effect of a particular size, but it does not establish that the effect exists. A computed power value is conditional on the model, design quantities, and specified alternative. In particular, it is not a probability that \(H_a\) is true after observing data.