Chance a batch of bags averages too heavy

Question

A machine fills bags to a true mean of 500 grams with standard deviation \sigma = 24 grams. Fills are independent, so by the central limit theorem the average of a batch is approximately normal. Across a batch of n = 36 bags, what is the probability the average weight exceeds 506 grams?

Accepted Answer

The batch average is approximately normal, centered at 500 g, with spread given by the standard error: 	ext{SE} = \frac{\sigma}{\sqrt{n}} = \frac{24}{\sqrt{36}} = \frac{24}{6} = 4 	ext{ grams}. Express the 506 g threshold in standard-error units (the z-score): z = \frac{506 - 500}{4} = \frac{6}{4} = 1.5. From the normal table: P(Z 1.5) \approx 0.0668, about 6.7\%. Even though single bags vary by 24 g, a 36-bag average landing 6 g high is a modestly rare event. Takeaway. The batch average has spread \sigma/\sqrt{n}: convert the threshold to a z-score, then read the normal tail. Here a 6 g ov…

Chance a batch of bags averages too heavy

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