A hospital administrator hoping to improve wait times decides to estimate the average emergency room waiting time at her hospital. She collects a simple random sample of 64 patients and determines the time (in minutes) between when they checked in to the ER until they were first seen by a doctor. A 95% confidence interval based on this sample is between 125 minutes and 148 minutes. Which of the following statements is true?

Question

BlackTom AI · Accepted Answer

The question asks us to interpret a 95% confidence interval for the population mean based on a simple random sample of n = 64.
Option 1: 'We are 95% confident that the average waiting time of all patients at this hospital’s emergency room is between 125 and 148 minutes.' This correctly reflects the meaning of a 95% confidence interval for the population mean, since the interval targets the true average waiting time for all patients, not just the sample.
Option 2: 'We are 95% confident that the average waiting time of these 64 emergency room patients is between 125 and 148 minutes.' This is inaccurate because the CI refers to the population mean, not the mean of this specific sample; the sample mean is a single value (and would be close to the endpoints in repeated sampling, but the interval describes the population parameter).
Option 3: '95% of random samples have a sample mean between 125 and 148 minutes.' This misstates the concept: the percentage refers to the population parameter, not the distribution of sample means, and it’s not correct to say that 95% of samples have their means inside that fixed interval.
Option 4: 'In order to decrease the margin of error of a 95% confidence interval to half of what it is now, we would need to double the sample size.' In fact, the margin of error is proportional to 1/√n, so halving the margin of error requires quadrupling the sample size, not merely doubling it.
Option 5: 'A 90% confidence interval would be wider than the 95% confidence interval.' Generally, a higher confidence level yields a wider interval; thus a 90% CI would be narrower, not wider, than a 95% CI.

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