Question: The yields from an ethanol-water distillation column have a standard deviation of 0.01. Process specifications call for a target yield of 0.93. Management wishes to detect any decrease in the true mean yield.

1. A random sample of 8 recent batches produced the following yields. Conduct a hypothesis test to determine whether the true mean yield has decreased. Use an \(\alpha=0.01 .\)
   
2. Based on the sample of 8 observations in part a, construct a \(99 \%\) confidence interval for the true mean yield of the process.
3. Discuss the relationship between your response to part a and part b.
4. What is \(\beta\), the probability of not rejecting the null hypothesis of part a, when in fact the true yield of the process is 0.925? What is \((1-\beta)\), the probability of rejecting the null hypothesis of part \(\mathrm{a}\), when in fact the true yield of the process is 0.925?

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