MULTIPLE LINEAR REGRESSION ANALYSIS A commercial bank wishes to determine the effectiveness of ATM's (Automatic Teller Machines) in reducing the service times for their customers. The bank takes a sam

MULTIPLE LINEAR REGRESSION ANALYSIS

A commercial bank wishes to determine the effectiveness of ATM's (Automatic Teller Machines) in reducing the service times for their customers. The bank takes a sample of 100 customers across several branches and collects data on the service time in seconds (Y) and the number of transactions (X1) that the customer completes. In this sample X1 varies from a low of 1 to a high of 6.

30 of the customers used an ATM and 70 of the customers used a human teller to carry out their transactions. Some of the customers were in urban branches whereas other customers were in branches outside of the urban area. To take this into account two additional variables were measured.

X2 = 1 if the customer uses an ATM

= 0 if the customer uses a human teller

X3 = 1 if the branch is in an urban area

= 0 if the branch is in an non-urban area.

The following four regressions were carried out with the following results:

I.    Yˆ = 40 + 100X1

II.  Yˆ = 62 + 90X1 - 38X2

III. Yˆ = 42 + 104X1 + 5X3

IV.  Yˆ = 68 + 87X1 - 34X2 + 2X3

Regression

Sum of Squares Due to Regression

Total Sum of Squares

I

50,000

70,000

II

66,740

70,000

III

52,000

70,000

IV

66,770

70,000

REGRESSION IV

VARIABLE

COEFFICIENT

STANDARD ERROR

Constant

68

5.6

X1

87

20

X2

-34

5.2

X3

2

2.1

Plots of the standardized residuals versus various variables for Regressions II and IV appeared to be random with no large outliers present.

1.   Compute the correlation coefficient between the number of transactions and the service time.

r-squared = SSregression/ sstotal  

                       =50,000/70,000  =0.714285714

          R =

2.   Write down the analysis of variance table for the regression of Y on X1, X2, and X3. Using it and any other information provided above, test for the statistical significance of the overall regression equation and determine which variables should be included and which variables should be deleted (if any). In your answer, state the null hypothesis and alternate hypothesis of all tests performed in terms of the Betas, compute the test statistics and explain how you arrive at your conclusions.

3.   Using the regression equation selected in question 2, compute the coefficient of determination and interpret.

4.   Estimate the service time for a customer who uses the ATM machine on 3 transactions in a non-urban branch and obtain a 95% prediction interval for this estimate.

5.   Michael, a customer in an urban branch, took 4.75 minutes (285 seconds) to complete 3 transactions at an ATM machine.

i.             What is the value of the residual if the simple linear regression model (model I) is used?

ii.            Compute the ratio of this residual to the standard error of estimate for the simple linear regression model.

iii.           Does this residual qualify as an outlier? Assuming that it does, explain why this happens.

6.   As the operations manager of this bank, you wish to evaluate the effectiveness of ATM machines. Write a report summarizing your conclusions. In your answer, use the results of the regression analysis to support your points numerically.

7.   A branch in a non-urban area processes 72,000 customer visits per year. Of these visits 36,000 are handled by human tellers and 36,000 are handled by ATM machines. To increase the use of ATM machines the bank adds more ATM machines and charges each customer $1 to use a human teller for each visit.  You are given the following cost and utilization data.

AutomaticTellerMachine (ATM)

Human Teller

Cost per hour

$15

$40

Average number of transactions per customer visit

2.40

1.80

Studies have shown that 40% of the customers will switch to an ATM machine from a human teller to avoid the service charge. Assuming that the average number of transactions per customer visit for those customers who switch to an ATM machine is the same as for those customers who choose not to switch, estimate the cost savings for this branch (1 hour = 3600 seconds).

NOTE: In your analysis on the cost savings, take into account the revenue obtained from the use of human tellers.