A 95% confidence interval for the average transaction amount at a retail store is calculated as [$45.20, $53.80]. Which of the following statements is true?

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BlackTom AI · Accepted Answer

Question restatement: A 95% confidence interval for the average transaction amount is [$45.20, $53.80]. Which statement is true?

Option 1: '95% of individual transaction amounts will fall within this interval.' This is incorrect because the interval targets the population mean, not every individual data point. Individual transaction amounts vary around the mean, and many will lie outside this interval.

Option 2: 'If repeated samples were taken, approximately 95% of the confidence intervals would contain the true mean.' This is the correct interpretation of a 95% confidence interval. It describes the long-run behavior: if we repeated the sampling process many times and constructed a CI from each sample, about 95% of those intervals would capture the true population mean.

Option 3: 'This interval guarantees that the sample mean is within $8.60 of the population mean.' Note that the full interval width is $8.60, but the statement about guaranteeing the sample mean being within $8.60 of the population mean is misguided. The CI provides a range for the population mean, not a guarantee about the sample mean relative to the population mean in every sample, and the notion of a fixed bound on the distance between means does not apply in this way.

Option 4: 'The probability that the true population mean is between $45.20 and $53.80 is exactly 95%.' This reflects a common misinterpretation. After the data are observed, the population mean is a fixed value, not a random variable, so the probability statement about the population mean is not correct. The 95% refers to the long-run proportion of intervals that would contain the mean across repeated sampling, not the probability that the true mean lies in this particular fixed interval.

A 95% confidence interval for the average transaction amount at a retail store is calculated as [$45.20, $53.80]. Which of the following statements is true?多项选择题

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