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A snack company sells bags of potato chips labeled with a net weight of \(200\) g. A consumer protection agency suspects the bags are underfilled.
They weigh a random sample of 50 bags and find a mean weight of \(196\) g with a standard deviation of \(8\) g.
  1. Construct a \(95\pourcent\) confidence interval for the true mean weight of the bags.
  2. Based on this interval, is the company's claim of \(200\) g valid? Justify your answer.

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