Question

10. (16 pts) The mean waiting time at the drive-through of a fast-food restaurant was 84.3 seconds last year. The manager wants to know whether the mean waiting time is reduced. She randomly selects a sample of customers and their waiting times are as follows:

108.5     67.4        58.0        75.9        65.1        80.4        95.5        86.3        70.9        72.0

1.     What is the population? What is the sample?
2.     At the α = 0.05 level of significance, determine whether there is evidence that the population mean is less than 84.3 seconds.
3.      Identify the type of error for your conclusion. Find the probability of error and probability of confidence if it is type I error.

Answer 1

(a) Population is the waiting time in seconds of all the patrons at the restaurant. Sample is the waiting time in seconds of the 10 randomly selected patrons.

(b)

Data:    

n = 10   

μ = 84.3   

s = 15.205   

x-bar = 78   

Hypotheses:    

Ho: μ ≥ 84.3   

Ha: μ < 84.3   

Decision Rule:    

α = 0.05   

Degrees of freedom = 10 - 1 = 9

Critical t- score = -1.833112923   

Reject Ho if t < -1.833112923   

Test Statistic:    

SE = s/√n = 15.205/√10 = 4.808243182

t = (x-bar - μ)/SE = (78 - 84.3)/4.80824318228602 = -1.310249869

p- value = 0.111281309   

Decision (in terms of the hypotheses):    

Since -1.310249869 > -1.833112923 we fail to reject Ho

Conclusion (in terms of the problem):

There is no sufficient evidence that the mean waiting time has reduced from the existing 84.3 seconds

(c) Probability of type I error = α = 0.05. The confidence level is 95%.