Question

(10 pts) The mean waiting time at the drive-through of a fast-food restaurant from the time the food is ordered to when it is received is 85 seconds. A manager devises a new system that he believes will decrease the wait time. He implements the new and measures the wait time for 10 randomly sampled orders. They are provided below: system 109 67 58 76 65 80 96 86 71 72 Assume the population is normally distributed. (a) Calculate the mean and standard deviation of the wait times for the 10 orders. (b) Construct a 99% confidence interval for the mean waiting time of the new system. (c) Given α-.05, determine whether the new systern is effective, ie, test whether the mean wait time is less than 85 seconds. Set up appropriate hypotheses, calculate your test statistic, find the rejection region and state your conclusion

Answer 1

(a)

Mean wait time = (109 + 67 + 58 + 76 + 65 + 80 + 96 + 86 + 71 + 72) / 10 = 78 seconds

Sample standard deviation, s = [(109 -78)² + (67 -78)² + (58 -78)² + (76 -78)² + (65 -78)² + (80 -78)² + (96 -78)² + (86 -78)² + (71 -78)² + (72 -78)² ] / 9 = 15.3912

(b)

Standard error of sampling distribution of mean = s / $\sqrt{n}$ = 15.3912 / $\sqrt{10}$ = 4.867125

Degree of freedom = n - 1 = 10 - 1 = 9

Critical value of t at df = 9 and 99% confidence interval is 3.25

99% confidence interval is,

(78 - 3.25 * 4.867125, 78 + 3.25 * 4.867125)

(62.18184, 93.81816)

(c)

Null hypothesis H0: Population mean wait time $\mu$ = 85 seconds

Alternative hypothesis Ha: Population mean wait time $\mu$ < 85 seconds

Test statistic = (Observed Mean - Hypothesized mean) / Standard error = (78 - 85) / 4.867125 = -1.438

Critical value of t at df = 9 and $\alpha$ = 0.05 is -1.83

Rejection region is test statistic < -1.83. That is, we reject H0, if test statistic is less than -1.83.

As, the observed test statistic (-1.438) is not in the rejection region, we fail to reject H0 and conclude that there is no significant evidence that the mean wait time is less than 85 seconds.