4.13 Waiting at an ER, Part I. 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 (128 minutes, 147 minutes), which is based on the normal model for the mean. Determine whether the following statements are true or false, and explain your reasoning.

  1. This confidence interval is not valid since we do not know if the population distribution of the ER wait times is nearly Normal.

False. Since n=64>30, the skew doesn’t matter as much and we can assume the confidence interval is valid.

  1. We are 95% confident that the average waiting time of these 64 emergency room patients is between 128 and 147 minutes.

False. We are 100% confident that the average waiting time is midway between 128 and 147 minutes. We are 95% confident that the average waiting time for the population at large is bewtween 128 and 147 minutes.

  1. We are 95% confident that the average waiting time of all patients at this hospital’s emergency room is between 128 and 147 minutes.

True.

  1. 95% of random samples have a sample mean between 128 and 147 minutes.

False, the confidence interval is not about the sample mean.

  1. A 99% confidence interval would be narrower than the 95% confidence interval since we need to be more sure of our estimate.

False. A 99% confidence interval would be wider than the 95% confidence interval. By including more options, we become more confident that we capture the true parameter.

  1. The margin of error is 9.5 and the sample mean is 137.5.

True. To find the margin of error, we can divide the distance of the interval by 2. (147-128)/2=19/2=9.5 and the sample mean is the midway point of the interval, so 128+9.5=137.5

  1. 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.

False. To half the margin of error, we would need to multiply the initial sample by (2^2)=4, since in the calculation of the SE we divide by the square root of the sample size.