Cereal boxes labeled 500 g, but the fill looks light

Question

A cereal brand labels each box as 500 g. The filling line's standard deviation is known to be $\sigma = 12$ g. A regulator weighs 64 boxes and finds a mean of 497 g.

Test at the 5% level (two-sided) whether the true mean fill differs from 500 g, and report the z-statistic.

Accepted Answer

Known \sigma, so a z-test applies. Hypotheses: H0: \mu = 500 versus H1: \mu 
eq 500. Standard error: 	ext{SE} = \frac{12}{\sqrt{64}} = \frac{12}{8} = 1.5 	ext{ g}. Test statistic: z = \frac{497 - 500}{1.5} = \frac{-3}{1.5} = -2.0. The critical values are \pm 1.96. Since -2.0 = 2.0 1.96, reject H0 (just barely). The two-sided p-value is 2\,[1 - \Phi(2.0)] = 2(0.0228) \approx 0.046. Conclusion: there is (marginal) evidence the boxes are underfilled. Takeaway. A z of exactly 2.0 sits just past the 1.96 line, so the verdict is "reject," but only barely: results this close to the boundary are f…

Cereal boxes labeled 500 g, but the fill looks light

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