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A cell phone company is thinking about opening a store in Millersburg. To help the company make a decision, its marketing department conducted a survey about cell phone ownership in the city. A total of $875$ randomly chosen Millersburg households participated in the survey. From the survey results, market researchers at the phone company calculated a $95\%$ confidence interval of for the mean number of cell phones owned by households in Millersburg. $\newline$ Is the following conclusion valid? $\newline$ If $100$ more surveys are conducted (each using a sample with members 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 $\newline$

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Interpret confidence intervals for population means

Full solution

Q. A cell phone company is thinking about opening a store in Millersburg. To help the company make a decision, its marketing department conducted a survey about cell phone ownership in the city. A total of

875

randomly chosen Millersburg households participated in the survey. From the survey results, market researchers at the phone company calculated a

95\%

confidence interval of for the mean number of cell phones owned by households in Millersburg.

\newline

Is the following conclusion valid?

\newline

If

100

more surveys are conducted (each using a sample with members 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

\newline

Understand Confidence Interval Definition: Understand what a $95\%$ confidence interval means. It indicates that if we were to take many samples and build a confidence interval from each of them, $95\%$ of these intervals would contain the true population mean.
Analyze Given Conclusion: Analyze the conclusion given. The statement suggests that all $100$ additional surveys will produce a confidence interval containing the sample mean, which is a misunderstanding of confidence intervals.
Recognize Incorrect Interpretation: Recognize that the statement is incorrect. The correct interpretation is that we expect about $95$ out of $100$ confidence intervals to contain the population mean, not the sample mean, and not all $100$ intervals.
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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