A large, national grocery retailer tracks the productivity and costs of its facilities closely. Data below were obtained from a single distribution center for one year. Each data point for each variable represents one week of activity. The variables included are the total labor hours (LabHours, Y), the number of cases shipped (Cases, in thousands, X₁), the indirect costs of the total labor hours as a percentage (Indirect%, X₂), a qualitative predictor called holiday that is coded as 1 if the week has a holiday and 0 otherwise (Holiday, X₃). The first 10 if the n = 54 cases are shown below, along with some multiple regression output:

a) Conduct a test for the overall significance of the model at thea = 0.01 level of statistical significance. State the null and alternative hypotheses, give the p-value of the test. What do you conclude? (3 points)

b) Based on the output, one of the data scientists running the analysis suggests that the model is better off without Indirect% as a measure. The data scientist proposes dropping out the variable from the model. Give two reasons why this may not be an appropriate step at this stage. (3 points)

c) Suppose now that an interaction term, Cases x Holiday, is added to the model and the resulting estimated multiple regression equation is:

Y-hat = 4251.778 + 0.823(Cases) - 12.157 (Indirect%) + 558.249 (Holiday) + 98.441 (Cases Holiday)

Give the slope of Cases (3 points):

(1) When a week has a holiday

(2) When a week does not have a holiday