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A manager at the Mega Raisin Cereal company wants to confirm that there is an acceptable number of raisins in the company's family-sized boxes. He counted the number of raisins in 100100 randomly sampled boxes from the Mega Raisin factory. From the collected data, he computed a 90%90\% confidence interval of for the mean number of raisins in family-sized boxes of Mega Raisin Cereal.\newlineIs the following conclusion valid?\newlineIf 100100 more samples are taken (with elements chosen randomly and independently), it is expected that exactly 100100 of them will each produce a 90%90\% confidence interval that contains its sample mean.\newlineChoices:\newline(A)yes\newline(B)no\newline

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Q. A manager at the Mega Raisin Cereal company wants to confirm that there is an acceptable number of raisins in the company's family-sized boxes. He counted the number of raisins in 100100 randomly sampled boxes from the Mega Raisin factory. From the collected data, he computed a 90%90\% confidence interval of for the mean number of raisins in family-sized boxes of Mega Raisin Cereal.\newlineIs the following conclusion valid?\newlineIf 100100 more samples are taken (with elements chosen randomly and independently), it is expected that exactly 100100 of them will each produce a 90%90\% confidence interval that contains its sample mean.\newlineChoices:\newline(A)yes\newline(B)no\newline
  1. Nature of Confidence Intervals: The statement is about the nature of confidence intervals. A 90%90\% confidence interval means that if we were to take many samples and compute a confidence interval for each sample, we would expect about 90%90\% of those intervals to contain the true population mean, not necessarily the sample mean.
  2. Incorrect Conclusion: The conclusion that exactly 100100 out of 100100 new samples will produce a 90%90\% confidence interval containing its sample mean is incorrect. The correct interpretation is that 9090 out of 100100 intervals would contain the population mean, not the sample mean.
  3. Correct Interpretation: Therefore, the correct answer is (B) no, because the conclusion misinterprets what a 90%90\% confidence interval represents.

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