A hospital administrator hoping to improve wait times decides to estimate the av

Question

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.

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

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

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

(d) 95% of random samples have a sample mean between 128 and 147 minutes.

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

(f) The margin of error is 9.5 and the sample mean is 137.5.

(g) 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.

Explanation / Answer

Answer:

(a) False. Provided the data distribution is not very strongly skewed (n = 64 in this sample, so we can be slightly lenient with the skew), the sample mean will be nearly normal, allowing for the normal approximation described.

(b) False. Inference is made on the population parameter, not the point estimate. The point estimate is always in the confidence interval.

(c) True.

(d) False. The confidence interval is not about a sample mean.

(e) False. To be more confident that we capture the parameter, we need a wider interval. Think about needing a bigger net to be more sure of catching a fish in a murky lake.

(f) True. Optional explanation: This is true since the normal model was used to model the sample mean. The margin of error is half the width of the interval, and the sample mean is the midpoint of the interval.

(g) False. In the calculation of the standard error, we divide the standard deviation by the square root of the sample size. To cut the SE (or margin of error) in half, we would need to sample 22 = 4 times the number of people in the initial sample (and we must assume that the same standard deviation applies, for instance if the population variance is known).

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