Refer to Exercise 7.22, where, in \(n=81\) cases, the coffee machine needed to be refilled with beans after 225 cups with a standard deviation of 22 cups.

(a) Conduct a test of hypotheses with the intent of showing that the mean number of cups is greater than 218 cups. Take \(\alpha=0.01\).

(b) Based on your conclusion in part (a), what error could you have made? Explain in the context of the problem.

Data From Exercise 7.22

(c) In a long series of repeated experiments, what proportions of the respective confidence intervals contain the true mean? Explain your reasoning.

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7.22 A caf records that in n = 81 cases, the coffee beans for the coffee machine lasted an average of 225 cups with a standard deviation of 22 cups. (a) Obtain a 90% confidence interval for u, the pop- ulation mean number of cups before the coffee machine needs to be refilled with beans. (b) Does lie in your interval obtained in part (a)? Explain.