Question

Historically, the population average waiting time to check out of a supermarket has been 4 minutes. Recently, in an effort to reduce the waiting time, the supermarket experimented with a recommendation system that generates real-time information to management the on number of cashiers to staff. The system involves infrared cameras that measure the amount of body heat in the checkout area of the store. The data from the cameras are feed in to an analytical software system that determines how many customers are waiting in line. After reviewing the results of the analytical software system, management determines how many cashiers to staff from the existing employees in other service areas of the store. A test of the new recommendation was conducted on a sample of 100 customers, and their mean waiting time to check out was 3.10 minutes, with a sample standard deviation of 2.5 minutes.

a. At the 0.05 level of significance determine if there is evidence to conclude that the recommendation system helped management to significantly reduce customer waiting time.

b. What technique did you use to determine your answer in part a?

c. Give the details necessary to demonstrate the technique to an audience.

d. What assumptions did you make in part a?

Answer 1

(a) = 4, = 3.1, s = 2.5, n = 100, = 0.05

The Hypothesis:

H0: =

Ha: <

This is a Left Tailed Test.

The Test Statistic: Although the population standard deviation is unknown, n is very large.

The test statistic is given by the equation:

The p Value:    The p value for Z = -3.6, is; p value = 0.0002

The Critical Value:   The critical value (Left Tail) at = 0.05, Zcritical = -1.645

The Decision Rule:   If Zobserved is < -Zcritical, Then reject H0.

Also if P value is < , Then Reject H0.

The Decision:   Since Zobserved (-3.1) is < -Zcritical (-1.645), We Reject H0 .

Also since P value (0.0002) is < (0.05) , We Reject H0.

The Conclusion: There is sufficient evidence at the 95% significance level to conclude that the mean waiting time to checkout has reduced.

___________________________________________________

(b) The technique use is a one sample, one tailed z test for a large sample (population standard deviation unknown)

___________________________________________________

(c) Assumptions made were that

(i) The population from which the sample came is normal or approximately normal

(ii) The sampling was a SRS (Simple random sample)

(iii) The samples are independent of each other.

___________________________________________________