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A food magazine decided to test the accuracy of Chips-a-ton's slogan: ＂Two thousand chocolate chips in every bag.＂ Magazine employees counted the number of chocolate chips in $100$ randomly selected Chips-a-ton bags. The magazine found a $90\%$ confidence interval of for the mean number of chocolate chips in Chips-a-ton bags. $\newline$ Is the following conclusion valid? $\newline$ If $100$ more samples are taken (with elements chosen randomly and independently), it is expected that exactly $90$ of them will each produce a $90\%$ confidence interval that contains its sample mean. $\newline$ Choices: $\newline$ (A)yes $\newline$ (B)no $\newline$

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Interpret confidence intervals for population means

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Q. A food magazine decided to test the accuracy of Chips-a-ton's slogan: ＂Two thousand chocolate chips in every bag.＂ Magazine employees counted the number of chocolate chips in

100

randomly selected Chips-a-ton bags. The magazine found a

90\%

confidence interval of for the mean number of chocolate chips in Chips-a-ton bags.

\newline

Is the following conclusion valid?

\newline

If

100

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

90

of them will each produce a

90\%

confidence interval that contains its sample mean.

\newline

Choices:

\newline

(A)yes

\newline

(B)no

\newline

Definition of $90$ % Confidence Interval: A $90\%$ confidence interval means that if we were to take many samples and calculate the confidence interval for each, we would expect $90\%$ of those intervals to contain the true population mean, not necessarily the sample mean.
Misunderstanding of Confidence Intervals: The conclusion that exactly $90$ out of $100$ additional samples will produce a $90\%$ confidence interval containing their sample mean is a misunderstanding of confidence intervals.
Focus on Population Mean: Confidence intervals are about the population mean, not the sample mean. The statement is incorrect because it guarantees a specific outcome for a random process, which is not how confidence intervals work.

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