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A physical trainer is designing a fitness program for middle-aged men. She asked a random sample of $125$ men in their $40s$ to report how far they could jump in centimeters from a standstill. The trainer computed a $90\%$ confidence interval of for the mean jump distance for men in their $40s$ . $\newline$ Is the following conclusion valid? $\newline$ There is a $90\%$ chance that the mean long jump distance for men in their $40s$ is in the interval . $\newline$ Choices: $\newline$ (A)yes $\newline$ (B)no $\newline$

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Interpret confidence intervals for population means

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Q. A physical trainer is designing a fitness program for middle-aged men. She asked a random sample of

125

men in their

40s

to report how far they could jump in centimeters from a standstill. The trainer computed a

90\%

confidence interval of for the mean jump distance for men in their

40s

.

\newline

Is the following conclusion valid?

\newline

There is a

90\%

chance that the mean long jump distance for men in their

40s

is in the interval .

\newline

Choices:

\newline

(A)yes

\newline

(B)no

\newline

Meaning of Confidence Interval: The confidence interval gives us the range in which the true mean is likely to fall, not the probability of the mean being within that range.
Interpretation of $90\%$ Confidence Interval: A $90\%$ confidence interval means that if we were to take many samples and build a confidence interval from each of them, about $90\%$ of those intervals would contain the true mean.
Common Misunderstanding: The conclusion that there is a $90\%$ chance that the mean is in the interval is a common misunderstanding of confidence intervals.
Correct Interpretation: The correct interpretation is that we are $90\%$ confident that the interval contains the true mean, not that there is a $90\%$ chance of the mean being in the interval.
Invalid Conclusion: Therefore, the conclusion given in the question prompt is not valid.

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