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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 $100$ randomly selected gamers. Each gamer was asked to note his or her playthrough time. Ultrafun found a $95\%$ confidence interval of for the mean playthrough time for all gamers in minutes. $\newline$ Is the following conclusion valid? $\newline$ If $100$ more samples are taken (with elements chosen randomly and independently), it is expected that exactly $100$ of them will each produce a $95\%$ confidence interval that contains its sample mean. $\newline$ Choices: $\newline$ (A)yes $\newline$ (B)no

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Interpret confidence intervals for population means

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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

100

randomly selected gamers. Each gamer was asked to note his or her playthrough time. Ultrafun found a

95\%

confidence interval of for the mean playthrough time for all gamers in minutes.

\newline

Is the following conclusion valid?

\newline

If

100

more samples are taken (with elements chosen randomly and independently), it is expected that exactly

100

of them will each produce a

95\%

confidence interval that contains its sample mean.

\newline

Choices:

\newline

(A)yes

\newline

(B)no

Definition of $95$ % Confidence Interval: A $95\%$ confidence interval means that if we were to take many samples and calculate the confidence interval for each, we would expect about $95\%$ of those intervals to contain the true population mean, not necessarily the sample mean.
Misconception about Confidence Intervals: The conclusion that exactly $100$ out of $100$ new samples will contain their sample mean within the $95\%$ confidence interval is incorrect. The confidence interval is about the population mean, not the sample mean. The statement misunderstands the concept of confidence intervals.
Correct Interpretation: The correct interpretation is that if we take many samples, we expect about $95\%$ 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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