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Gary is a statistician working for the Hancock County labor department. As part of a state-mandated data collection project, he distributed a survey about vacation and other employment benefits to $725$ randomly chosen residents of Hancock County. From the survey results, Gary calculated a $99\%$ confidence interval of for the mean number of vacation days Hancock County residents took last year. $\newline$ Is the following conclusion valid? $\newline$ There is a $99\%$ chance that the mean number of vacations days Hancock County residents took last year is in the interval . $\newline$ Choices: $\newline$ (A)yes $\newline$ (B)no

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Interpret confidence intervals for population means

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Q. Gary is a statistician working for the Hancock County labor department. As part of a state-mandated data collection project, he distributed a survey about vacation and other employment benefits to

725

randomly chosen residents of Hancock County. From the survey results, Gary calculated a

99\%

confidence interval of for the mean number of vacation days Hancock County residents took last year.

\newline

Is the following conclusion valid?

\newline

There is a

99\%

chance that the mean number of vacations days Hancock County residents took last year is in the interval .

\newline

Choices:

\newline

(A)yes

\newline

(B)no

Interpreting Confidence Intervals: A confidence interval does not give the probability of the parameter (like the mean number of vacation days) being in the interval. Instead, it means that if we were to take many samples and build a confidence interval from each of them, $99\%$ of those intervals would contain the true mean.
Common Misconception: The statement ＂There is a $99\%$ chance that the mean number of vacations days Hancock County residents took last year is in the interval＂ is a common misconception. The correct interpretation is that we are $99\%$ confident that the interval captures the true mean number of vacation days.
Validity of Conclusion: The conclusion is not valid because it misinterprets the meaning of a confidence interval. The correct interpretation is about the confidence in the method used to create the interval, not the probability of the mean being in the interval.

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