Hypothesis Test - Emergency Room Waiting Times

Hypothesis Test - Emergency Room Waiting Times

A hospital administrator is concerned about the waiting times for patients in the emergency room. She records the waiting times (in minutes) of a random sample of 32 patients, which are shown below:

133	263	99	192	401	318	202	120	136	195	167	237	89	238	186	137
256	172	333	210	158	74	165	321	219	124	203	371	136	160	81	192

From these data, the sample mean is calculated to be 196.5 minutes. Suppose the standard deviation of all waiting times in this emergency room is known to be 78.4 minutes. (a) We have no knowledge about whether waiting times follow a normal distribution. Why is it nevertheless appropriate to use inference methods which rely on the assumption of normality?

The Central Limit Theorem (CLT) says that if the sample size is large then the distribution of the sample mean follows a normal distribution regardless the distribution of the data. Now in statistics a large sample implies a sample with size more than 30. As here the sample size is more than 30 so we can use CLT and thus assumption of normality is valid here. (b) Construct a 97% confidence interval for the true mean emergency waiting time for this hospital. Explain how you find the critical value.

As here we need 95% confidence interval so we need 3% significance level thus the critical value is,

Z(0.03/2) = Z(0.015) = 2.17 from normal table.

From the data,

Sample mean = 196.5 and sample standard deviation =83.1.

So Standard error of the sample mean =

97% CI = (Sample mean ±Critical value*SE)

= (196.5 ± 2.17*14.69) = (164.623, 228.377) (c) Provide an interpretation of the interval in (b).

The interpretation is if we randomly select large number of samples and create the above Confidence Interval then at least 97% of them will contain the true mean value. (d) Conduct a hypothesis test at the 3% level of significance to determine whether the true mean emergency room waiting time differs from 3 hours (i.e., 180 minutes). Show all of your steps, including the hypotheses, the calculation of the test statistic and P-value, and a properly worded conclusion.

Hypothesis:

Critical value = ±Z(0.015) = ±2.17

Rejection condition: Observed | Z|>Critical value = ±Z(0.015) = ±2.17

Z =

P-value = P( |Z| > 1.123) = 0.2614

As test statistic does not lie in the critical region so we are failing to reject the null hypothesis. So at 97% confidence the true mean emergency room waiting time does not differ from 3 hours (i.e., 180 minutes). (e) Interpret the meaning of the P-value you calculated in (d).

The p-value 0.2614 means that under the null hypothesis we have 0.2614 probability of getting the value of the test statistic 1.123 or extreme.

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