Chez Henri is a restaurant chain that operates in 40 different cities. It hired an economist

Chez Henri is a restaurant chain that operates in 40 different cities. It hired an economist to estimate the factors affecting the demand for its sales. The following equation was estimated using cross sectional data from each of its 40 restaurants.

Y Annual restaurant sales (in thousands) X1 Disposable per capital income (in thousands) of the residents living within 5

miles of a restaurant X2 Population (in thousands) within a 5-mile radius of a restaurant X3 Number of competing restaurants within a 5-mile radius

The following information was obtained from the regression analysis: Multiple R: 0.92 R-Square: 0.85 Std. Error of Est.: 0.40

Analysis of Variance

DF Sum Squares Mean Sqr. F-Stat Regression 3 220 73.3 18.2 Residual 36 60 1.7 Variable Coefficient Std. Error T-Value Constant 0.4 0.2 2.0 X1 0.01 0.004 2.5 X2 0.02 0.015 1.3 X3 -20.2 4.50 -4.6

Answer the following questions: a. Give the estimated demand equation for predicting restaurant sales.

The estimated demand equation is, Annual restaurant sales (in thousands)= 0.4 + 0.01* Disposable per capital income (in thousands)+ 0.02* Population (in thousands) – 20.2* Number of competing restaurants

b. Provide an interpretation for each of the regression coefficients.

The regression coefficients tell us how the sales will change if the value of the variable changes per unit provided the value of the other variable is fixed. As for example the coefficient 0.01 which is associated with “Disposable per capital income (in thousands) of the residents living within 5 miles of a restaurant” implies that if the other factors are fixed then per unit($1000 per capita income) increase(decrease) will lead to 0.01 unit (unit in $1000) increase(decrease) in annual sales. The coefficient associated with constant implies that if the values of the other variables are zero then the annual sale would be $400.

c. Which of the coefficients are statistically significant and which are not? Explain.

We can answer that by looking at the corresponding t values. Here the t-statistics follows a t distribution with degrees of freedom = 36. And for that d.f the critical value is 2.339 at 5% significance level (more precisely the critical value are -2.339, 2.339). And we can

see the t-stat value corresponding to X1 and X3 are in the critical regions thus these two variables are statistically significant and rest are not.

d. What percent of variation are restaurant sales explained by this equation?

The coefficient of determination or R-sq tells us what percent of variation of restaurant sales are explained by this equation. Here the value 0.85 is telling us that 85% of the variation is explained.