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A major insurance company wants to estimate the effort it would take to completely switch to paper-free correspondence. The company randomly selected $300$ customers and logged the number of letters sent to each last month. They found a $95\%$ confidence interval of for the mean number of letters sent to each customer last month. $\newline$ Is the following conclusion valid? $\newline$ If the company takes another random sample, there is a $95\%$ chance that the mean number of letters sent to each customer last month will be in the new sample's $95\%$ confidence interval. $\newline$ Choices: $\newline$ (A)yes $\newline$ (B)no

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Interpret confidence intervals for population means

Full solution

Q. A major insurance company wants to estimate the effort it would take to completely switch to paper-free correspondence. The company randomly selected

300

customers and logged the number of letters sent to each last month. They found a

95\%

confidence interval of for the mean number of letters sent to each customer last month.

\newline

Is the following conclusion valid?

\newline

If the company takes another random sample, there is a

95\%

chance that the mean number of letters sent to each customer last month will be in the new sample's

95\%

confidence interval.

\newline

Choices:

\newline

(A)yes

\newline

(B)no

Understand Confidence Interval: Understand the concept of a confidence interval. A $95\%$ confidence interval means that if we were to take many samples and build a confidence interval from each of them, $95\%$ of those intervals would contain the true mean.
Analyze Conclusion: Analyze the given conclusion. $\newline$ The conclusion suggests that a new sample's mean will fall within the previously calculated confidence interval $95\%$ of the time, which is a misunderstanding of what a confidence interval represents.
Identify Error: Identify the error in reasoning. $\newline$ The correct interpretation is that $95\%$ of such confidence intervals from repeated random samples will contain the true population mean, not that a new sample's mean will fall within a specific previously calculated interval.
Choose Correct Answer: Choose the correct answer based on the analysis. $\newline$ The conclusion given is not valid, so the answer is $(B)$ no.

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