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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 875875 randomly chosen Millersburg households participated in the survey. From the survey results, market researchers at the phone company calculated a 95%95\% confidence interval of for the mean number of cell phones owned by households in Millersburg.\newlineIs the following conclusion valid?\newlineIf 100100 more surveys are conducted (each using a sample with members chosen randomly and independently), it is expected that exactly 100100 of them will each produce a 95%95\% confidence interval that contains its sample mean.\newlineChoices:\newline(A)yes\newline(B)no\newline

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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 875875 randomly chosen Millersburg households participated in the survey. From the survey results, market researchers at the phone company calculated a 95%95\% confidence interval of for the mean number of cell phones owned by households in Millersburg.\newlineIs the following conclusion valid?\newlineIf 100100 more surveys are conducted (each using a sample with members chosen randomly and independently), it is expected that exactly 100100 of them will each produce a 95%95\% confidence interval that contains its sample mean.\newlineChoices:\newline(A)yes\newline(B)no\newline
  1. Understand Confidence Interval Definition: Understand what a 95%95\% confidence interval means. It indicates that if we were to take many samples and build a confidence interval from each of them, 95%95\% of these intervals would contain the true population mean.
  2. Analyze Given Conclusion: Analyze the conclusion given. The statement suggests that all 100100 additional surveys will produce a confidence interval containing the sample mean, which is a misunderstanding of confidence intervals.
  3. Recognize Incorrect Interpretation: Recognize that the statement is incorrect. The correct interpretation is that we expect about 9595 out of 100100 confidence intervals to contain the population mean, not the sample mean, and not all 100100 intervals.
  4. Choose Correct Answer: Choose the correct answer based on the analysis. The conclusion is not valid because it misinterprets the meaning of a 95%95\% confidence interval.

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