A new machine. Are the loaves lighter?

One loaf proves nothing: loaves vary. The mean of a sample varies far less, and that is what makes it evidence.

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Loaf masses have been $N(800,\ 15^2)$ in grams: the wide dashed curve. A single loaf of 794 g is only 0.4 of a standard deviation light. Nothing to see.

The mean of a sample of $n$ has variance $\tfrac{15^2}{n}$. With 25 loaves,

a much narrower curve: averaging cancels most of the variation.

Test $H_0: \mu = 800$ against $H_1: \mu < 800$ at 5%. Assuming $H_0$, the lowest 5% of sample means lie below

The sample mean, 794 g, lands inside the 5% region: reject $H_0$. But at 1% the line moves to $800 - 2.3263 \times 3 = 793.02$ and 794 is no longer beyond it. The stricter test needs stronger evidence.

Test the mean with the mean's own curve.

One- and two-tailed tests, critical regions and p-values, choosing a sample size, and twelve graded exam questions with full worked solutions are waiting.

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