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?
(b) Construct a 97% confidence interval for the true mean emergency waiting time for this hospital. Explain how you find the critical value.
(c) Provide an interpretation of the interval in (b).
(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.
(e) Interpret the meaning of the P-value you calculated in (d).
(g) Could the interval in (b) have been used to conduct the test in (d)? If no, explain why not. If yes, explain why, and explain what your conclusion would have been and why.