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During archery practice one day, Sadie asked a friend to take a picture of the target after she shot each arrow. After practice, they measured the distance from the bull's-eye (in centimeters) in $100$ randomly selected pictures. They found a $99\%$ confidence interval of for the mean distance from the bull's eye for the arrows Sadie shot that day. $\newline$ Is the following conclusion valid? $\newline$ If $100$ more samples are taken (with elements chosen randomly and independently), it is expected that exactly $99$ of them will each produce a $99\%$ 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. During archery practice one day, Sadie asked a friend to take a picture of the target after she shot each arrow. After practice, they measured the distance from the bull's-eye (in centimeters) in

100

randomly selected pictures. They found a

99\%

confidence interval of for the mean distance from the bull's eye for the arrows Sadie shot that day.

\newline

Is the following conclusion valid?

\newline

If

100

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

99

of them will each produce a

99\%

confidence interval that contains its sample mean.

\newline

Choices:

\newline

(A)yes

\newline

(B)no

Confidence Interval Definition: A $99\%$ confidence interval means that we can be $99\%$ confident that the true population mean lies within the interval, not that $99$ out of $100$ samples will contain the sample mean within their own confidence intervals.
Common Misunderstanding: The statement is a common misunderstanding of confidence intervals. The correct interpretation is that if we were to take many samples and build a confidence interval from each, $99\%$ of those intervals would contain the true population mean, not necessarily the sample mean.
Correct Interpretation: Therefore, the conclusion that exactly $99$ of the $100$ new samples will produce a $99\%$ confidence interval containing its sample mean is incorrect.

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