Level 4: Inference · Problem 63 of 87

Power of a one-sided z test

A statistics problem from Statisticsdrill. Solve it with one answer box, with a fresh sample on every attempt and the worked steps and spreadsheet function in the answer, and it is checked the moment you submit.

The task

A bottling line is supposed to fill bottles to a mean of 488 ml, and the process standard deviation is known to be σ = 7. A sample of n = 49 will be used to test H0: μ = 488 against H1: μ < 488 with a z test at α = 0.1. If the true mean is actually 485.5 ml, what is the power of the test? Answers within 0.004 count, so z values rounded to 2 decimals work. Round to 4 decimal places.

What this problem teaches

The formula: Power = Φ( |μ1 - μ0| √n / σ - z_α ). Power rises with a bigger sample, a bigger true shift, a smaller σ, or a larger α. It is 1 - β, where β is the chance of missing a real effect.

A hint if you get stuck

Power = P(reject H0 | true mean). Reject when z = (x̄ - μ0) / (σ/√n) is beyond z_α. Shift the cutoff to the true mean: power = Φ(|μ1 - μ0| √n / σ - z_α).