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A lighting company is estimating the life span of its new LED lightbulb. Company engineers randomly selected 775775 lightbulbs from the first product shipment for testing and recorded the number of hours each selected lightbulb lasted. From the test results, the company estimated a 95%95\% confidence interval of for the mean life span of lightbulbs in the first shipment.\newlineIs the following conclusion valid?\newlineIf 100100 more samples are taken (with elements chosen randomly and independently), it is expected that exactly 9595 of them will each produce a 95%95\% confidence interval that contains its sample mean.\newlineChoices:\newline(A)yes\newline(B)no

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Q. A lighting company is estimating the life span of its new LED lightbulb. Company engineers randomly selected 775775 lightbulbs from the first product shipment for testing and recorded the number of hours each selected lightbulb lasted. From the test results, the company estimated a 95%95\% confidence interval of for the mean life span of lightbulbs in the first shipment.\newlineIs the following conclusion valid?\newlineIf 100100 more samples are taken (with elements chosen randomly and independently), it is expected that exactly 9595 of them will each produce a 95%95\% confidence interval that contains its sample mean.\newlineChoices:\newline(A)yes\newline(B)no
  1. Definition of 9595% Confidence Interval: A 95%95\% confidence interval means that if we were to take many samples and calculate a confidence interval for each, we would expect 95%95\% of those intervals to contain the true population mean.
  2. Misunderstanding of Confidence Intervals: The conclusion states that if 100100 more samples are taken, exactly 9595 of them will produce a 95%95\% confidence interval that contains the sample mean. This is a misunderstanding of confidence intervals.
  3. Focus on Population Mean: Confidence intervals are about the population mean, not the sample mean. The statement should refer to the population mean being within the confidence intervals, not the sample means.
  4. Correct Interpretation: The correct interpretation is that we expect about 9595 of the 100100 confidence intervals to contain the true population mean, not exactly 9595.
  5. Invalid Conclusion: Therefore, the conclusion given in the problem is not valid because it misinterprets the meaning of a 95%95\% confidence interval.

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