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Simple Linear Regression Equation Part 2
In the Part 1, we have received linear regression equation for the following problem:
The firm has decided to use linear regression by employing the equation y = a + bx for the annual sales. The prior years's data regarding sales and advertising expenditures are here.
Linear regression equation is
y = f(x) = 4.2 + 0.31x
Let y' is the mean of the observed data points, i.e.
y' = (y1 + y2 + y3 + y4 + y5)/5
1. The variance of the estimate S2 is equal:
where n = 5 is the number of observations, k = 1 is the number of independent variables in the linear regression. n-k-1 is calling the number of degrees of freedom.
Hence, S2 = 2.624
The standard error of estimate is equal square root of the variance of the estimate and is equal S. Hence, standard error of estimate S = 1.62
2. Variance of constant coefficient a is equal Sa2:
Sa2 = 6.216
Hence, Sa - standard error of a is equal square root of the variance of a and is equal Sa = 2.49
3. Similarly, variance of variable coefficient b is equal Sb2:
Sb2 = 0.002332
Hence, Sb - standard error of b is equal square root of the variance of b and is equal Sb = 0.0483
4. Coefficient of determination r2:
r2 = 0.9328
r - coefficient of correlation, r = 0.9658
The coefficient of determination r2 may be interpreted as the percent of variation in the depended variable "explained" by the variation in the independent variable. Hence, the variance in annual sales is 93.28% of the variance in advertising cost.
5. Confidence interval for b (similarly for other parameters):
If we want to receive 90% confidence interval for b, then we need the t-value from Student's t-table: t(5-1-1,(1-0.90)/2) = t(3 , 0.05) =2.35
Hence, management may be 90% confident that b will be in the range of:
(0.1965; 0.4235)
The two-third confidence level falls within one standard deviation (approximately) from the mean. So, if advertising expenditures will be $40 000 (40 thousands) management can be two-third confident that the annual sales will be in the range of:
( 4.2+0.31*40 – 1.62; 4.2+0.31*40 + 1.62)
or
(14.98; 18.22)
or
($14 980 000; $18 220 000).
Posted by mazoo at March 21, 2005 9:25 PM
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Simple Linear Regression Equation Part 1 Mar 03, 2005
Comments
thanks for excellent expression
Posted by: cuijf at February 22, 2008 8:33 AM