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Education officials in Fairview are considering adjustments to the community service requirements for high school graduation. They conducted a survey to assess the typical involvement in volunteering among local high school students. In the survey, $75$ randomly chosen Fairview students reported their volunteer hours from the last year. The officials calculated a $95\%$ confidence interval of for the mean number of hours Fairview students volunteered last year. $\newline$ Is the following conclusion valid? $\newline$ If the officials conduct another survey, there is a $95\%$ chance that the mean number of hours Fairview students volunteered last year will be in the new survey's $95\%$ confidence interval. $\newline$ Choices: $\newline$ (A)yes $\newline$ (B)no

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Interpret confidence intervals for population means

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Q. Education officials in Fairview are considering adjustments to the community service requirements for high school graduation. They conducted a survey to assess the typical involvement in volunteering among local high school students. In the survey,

75

randomly chosen Fairview students reported their volunteer hours from the last year. The officials calculated a

95\%

confidence interval of for the mean number of hours Fairview students volunteered last year.

\newline

Is the following conclusion valid?

\newline

If the officials conduct another survey, there is a

95\%

chance that the mean number of hours Fairview students volunteered last year will be in the new survey's

95\%

confidence interval.

\newline

Choices:

\newline

(A)yes

\newline

(B)no

Definition of 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 $95\%$ of those intervals to contain the true population mean.
Misunderstanding of $95\%$ Confidence Interval: The conclusion is suggesting that there's a $95\%$ chance the true mean falls within the interval calculated from a single new survey. This is a misunderstanding of confidence intervals.
Purpose of Confidence Intervals: Confidence intervals do not predict the outcome of future samples. They provide an estimate of where the true population parameter lies based on the sample data we currently have.
Correct Interpretation of Confidence Intervals: The correct interpretation is that we are $95\%$ confident that the true mean of the population falls within the interval we calculated, not that future sample means will fall within this interval.

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