Amanufacturer of detergent claims that the contents of boxes sold "are guaranteed to weigh on average at least 18 ounces". if this claim turns out to be false, the manufacturer will face a high fine. the distribution of weight is known to be normal, with a standard deviation of 0.6 ounce. a random sample of 16 boxes yielded a sample mean weight of 19ounces. (a)does the sample provide evidence at the 5% significance level to put the manufacturer's mind at peace in terms of the possible fine? (b)does the sample provide evidence at the 5% significance level for the state to persecute the manufacturer for false claim?

Given the sample mean of 19 oz, an SD of 0.6 oz, and a sample size of 16 boxes:
The bounds of the confidence interval are calculated as:
mean +/- z*SD/sqrt(n), where z = 1.65 for a one-tailed 95% CI
19 +/- 1.65*0.6/sqrt(16)
19 +/- 1.65*0.6/4
19 +/- 0.25
18.75 to 19.25
a) Since this entire interval is well above the 18.00 threshold, the manufacturer is statistically justified in making his claim.
b) The state does not have statistical evidence to claim that the manufacturer is making false claims about his product. This would only occur if the entire interval is below the threshold of 18.
In the case that 18 is included in the interval (for example, if the confidence interval is from 17.9 to 18.4), then neither side is able to claim anything with certainty, because there will not be enough evidence for the manufacturer to say that his samples are more than 18 oz, but neither will the state have sufficient evidence to claim that most samples are less than 18 oz.

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