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Drinkers of Jitterfuel iced tea complain that the amount of caffeine per bottle varies too much. Jitterfuel decided to test these claims by measuring the amount of caffeine in $125$ randomly selected bottles from the latest production batch. The company found a $95\%$ confidence interval of for the mean amount of caffeine in bottles from the production batch (in milligrams). $\newline$ Is the following conclusion valid? $\newline$ If $100$ more samples are taken (with elements chosen randomly and independently), it is expected that exactly $100$ of them will each produce a $95\%$ 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. Drinkers of Jitterfuel iced tea complain that the amount of caffeine per bottle varies too much. Jitterfuel decided to test these claims by measuring the amount of caffeine in

125

randomly selected bottles from the latest production batch. The company found a

95\%

confidence interval of for the mean amount of caffeine in bottles from the production batch (in milligrams).

\newline

Is the following conclusion valid?

\newline

If

100

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

100

of them will each produce a

95\%

confidence interval that contains its sample mean.

\newline

Choices:

\newline

(A)yes

\newline

(B)no

Understand Confidence Interval Definition: Understand what a $95\%$ confidence interval means. It implies that if we were to take many samples and calculate the confidence interval for each, about $95\%$ of these intervals would contain the true population mean.
Analyze Given Conclusion: Analyze the conclusion given. It states that if $100$ more samples are taken, exactly $100$ of them will produce a $95\%$ confidence interval that contains its sample mean. This is a misunderstanding of confidence intervals.
Recognize Error in Conclusion: Recognize the error in the conclusion. The correct interpretation is that we expect about $95$ out of $100$ samples to produce a confidence interval that contains the population mean, not the sample mean as stated.
Choose Correct Answer: Choose the correct answer based on the analysis. The conclusion is not valid because it misinterprets the meaning of a $95\%$ confidence interval.

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