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Managers at Ken's Sporting Goods are concerned that the dumbbells from one of the store's new suppliers are poorly made. To investigate, the managers used a precision scale to weigh a random sample of 5050 weights labeled ＂30 kg30\,\text{kg}.＂ They computed a 95%95\% confidence interval of for the mean weight of ＂30 kg30\,\text{kg}＂ dumbbells from their new supplier (in kilograms).\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 95%95\% confidence interval that contains its sample mean.\newlineChoices:\newline(A)yes\newline(B)no\newline

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Q. Managers at Ken's Sporting Goods are concerned that the dumbbells from one of the store's new suppliers are poorly made. To investigate, the managers used a precision scale to weigh a random sample of 5050 weights labeled ＂30 kg30\,\text{kg}.＂ They computed a 95%95\% confidence interval of for the mean weight of ＂30 kg30\,\text{kg}＂ dumbbells from their new supplier (in kilograms).\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 95%95\% confidence interval that contains its sample mean.\newlineChoices:\newline(A)yes\newline(B)no\newline
  1. Definition of 9595% Confidence Interval: A 95%95\% confidence interval means that if we were to take many samples and compute a confidence interval for each sample, we would expect about 95%95\% of those intervals to contain the true population mean.
  2. Misunderstanding of Confidence Intervals: The conclusion states that exactly 100100 out of 100100 additional samples will produce a 95%95\% confidence interval that contains its sample mean. This is a misunderstanding of confidence intervals.
  3. Correct Interpretation: The correct interpretation is that we expect about 9595 out of 100100 samples to produce a confidence interval that contains the true population mean, not necessarily the sample mean.
  4. Invalid Conclusion: Therefore, the conclusion that exactly 100100 of them will contain the sample mean is not valid because it misinterprets the meaning of a 95%95\% confidence interval.

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