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Game reviewers complained extensively about how Ultrafun gaming company's last game was too short. To ensure this does not happen again, Ultrafun gave a pre-release copy of its new game to 100100 randomly selected gamers. Each gamer was asked to note his or her playthrough time. Ultrafun found a 95%95\% confidence interval of for the mean playthrough time for all gamers in minutes.\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

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Q. Game reviewers complained extensively about how Ultrafun gaming company's last game was too short. To ensure this does not happen again, Ultrafun gave a pre-release copy of its new game to 100100 randomly selected gamers. Each gamer was asked to note his or her playthrough time. Ultrafun found a 95%95\% confidence interval of for the mean playthrough time for all gamers in minutes.\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
  1. Definition of 9595% Confidence Interval: A 95%95\% confidence interval means that if we were to take many samples and calculate the confidence interval for each, we would expect about 95%95\% of those intervals to contain the true population mean, not necessarily the sample mean.
  2. Misconception about Confidence Intervals: The conclusion that exactly 100100 out of 100100 new samples will contain their sample mean within the 95%95\% confidence interval is incorrect. The confidence interval is about the population mean, not the sample mean. The statement misunderstands the concept of confidence intervals.
  3. Correct Interpretation: The correct interpretation is that if we take many samples, we expect about 95%95\% of the confidence intervals calculated from those samples to contain the true population mean. It does not guarantee that every single confidence interval will contain its sample mean.

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