Question

Computers Galore, Inc., is trying to construct a consumer profile to know how to direct its marketing efforts. The store has data on twenty recent customers, including the amount spent on each customer's computer system (Ex), the customer's annual household income (M), and the number of years of education (Ed).

A linear regression of Ex on M and Ed yields the following results by the spreadsheet:

           

Regression Output:
	Estimates:	Std Error of Coefficients
Constant	‑1799.18
Income	0.008423	0.009895
Education	251.7695	71.98489
Std Err of Y Est	821.7464
R Squared	0.456706
Observations	20
Degrees of Freedom	17

a.	You are asked by the store manager to estimate how much a college educated person (16 years) earning $40,000 a year is likely to spend on a computer system at CGI. Give your best estimate and construct an approximately 95% confidence interval for your estimate.
b.	The store manager thinks both income and education are roughly equally important   determinants of how much a customer spends on a computer system. Do the results of the regression bear this out? Explain with reference to t statistics. (A t statistic is calculated by t = b/sb.)

Answer 1

b) Other things being equal, the value of a t test will be large in absolute terms whenever the value of the sample regression coefficient is large in absolute terms. However, the value of the t test also depends on the magnitude of the standard error of the sample regression coefficient. In short, the deviation of any sample regression coefficient from the population regression coefficient, which is typically assumed to be equal to zero, must be assessed in relation to the standard deviation of the sampling distribution of the regression coefficient. Even if the sample regression coefficient is large, the t test will not be large if the standard error of this coefficient is also large. Indeed, an examination of the t distributions for moderately large samples indicates that a sample regression coefficient must be roughly twice the magnitude of its standard error if it is to be statistically significant at the conventional 0.05 probability level.

The significance of a regression coefficient in a regression model is determined by dividing the estimated coefficient over the standard deviation of this estimate. For statistical significance we expect the absolute value of the t-ratio to be greater than 2 or the P-value to be less than the significance level (α=0,01 or 0,05 or 0,1).

now for

Income - t-statitic = 0.008423/ 0.009895 = 0.85123 < 2.Hence not statistically significant in explaining model.

education - t-statistic =  251.7695 / 71.98489 = 3.497 > 2 .Hence statistically significant in explaining model.