Question

Just the Fax, Inc. (JTF) hired a consultant to estimate the demand for its line of telecommunications devices in 35 different market areas for one month. The available data set includes observations on the number of thousands of units sold by JTF per month (QX), the price per unit charged by JTF (PX), the average unit price of competing brands (PZ), monthly advertising expenditures by JTF (A), and average gross sales (in thousands of dollars) of businesses in the market area (I). The consultant provided the results below without the estimated coefficients. However, she provided the following:

- Standard errors (in parenthesis) - t-stats [in square brackets]
- R square.

QX=b0 +b1PX +b2PZ +b3A+b4I (250) (1.4) (0.8) (0.05) (0.04) [1.5] [-2.5] [1.5] [3.0] [2.5]

R2 = 0.80

JTF has hired you as a second consultant to produce the estimated coefficients, and to determine whether they are statistically significant.
Answer the following questions using the table below:

i. Calculate the missing coefficients for the intercept and the four independent variables in column 2 of the table below.

ii. If the critical t value at 5% significance level is 2.04, indicate in column 3 whether the calculated coefficients are significantly different from zero.

Variable	Coefficient	Significant/Not Significant
Intercept
Px
Pz
A
I

iii. What is the interpretation of the R2 value of 0.80?

Answer 1

i)

Estimated value of bo=t-stat*Standard error=1.5*250=375

Estimated value of b₁=t-stat*Standard error=-2.5*1.4=-3.5

Estimated value of b₂=t-stat*Standard error=1.5*0.8=1.2

Estimated value of b₃=t-stat*Standard error=3*0.05=0.15

Estimated value of b₄=t-stat*Standard error=2.5*0.04=0.1

ii)

Absolute value of t for observed value of bo is 1.5 which is lower than critical value of t i.e. 2.05. We can say that intercept is not statistically significant.

Absolute value of t for observed value of b₁ is -2.5 which is higher than critical value of t i.e. 2.05. We can say that there is a significant correlation between Q_x and P_x. Observed value of b₁ is statistically significant.

Absolute value of t for observed value of b₂ is 1.5 which is lower than critical value of t i.e. 2.05. We can say that there is not a significant correlation between Q_x and P_Z. Observed value of b₂ is not statistically significant.

Absolute value of t for observed value of b₃ is 3 which is higher than critical value of t i.e. 2.05. We can say that there is a significant correlation between Q_x and A. Observed value of b₃ is statistically significant.

Absolute value of t for observed value of b₄ is 2.5 which is higher than critical value of t i.e. 2.05. We can say that there is a significant correlation between Q_x and I. Observed value of b₃ is statistically significant.

Variable	Coefficient	Significant/Not significant
Intercept	375	Not significant
Px	-3.5	Significant
PZ	1.2	Not significant
A	0.15	Significant
I	0.1	Significant

iii)

We get coefficient of determination equal to 0.80.

It means that the model explains 80% variability of the response data around its mean.