Power of a test to catch underfilled bags

Question

A filling machine is supposed to put 500 g in each bag. Quality control tests H0: \mu = 500 against H1: \mu < 500 (underfilling), using a known standard deviation \sigma = 8 g, a sample of n = 16 bags, and a one-sided test at \alpha = 0.05. Suppose the machine has actually drifted to a true mean of \mu = 495 g. What is the power of the test to detect this drift?

Accepted Answer

Standard error. \sigma/\sqrt{n} = 8/\sqrt{16} = 8/4 = 2 g. Cutoff. A lower-tail test at \alpha = 0.05 rejects when the sample mean falls below \mu0 - z\alpha \cdot 	ext{SE} = 500 - 1.645 	imes 2 = 496.71 g. Effect size in SE units. The true mean sits (500 - 495)/2 = 2.5 standard errors below the null. Power. The chance the sample mean lands below the cutoff when the true mean is 495: 	ext{power} = \Phi(	ext{effect} - z\alpha) = \Phi(2.5 - 1.645) = \Phi(0.855) \approx 0.80. A five-gram drift is large relative to the standard error, so with just 16 bags the test still catches it about 80% o…

Power of a test to catch underfilled bags

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