Question

7. Multiple regression analysis is used to study how an individual's income (y, in thousands of dollars) is influenced by age (x1, in years), level of education (22, ranging from 1 to 5), and the individual's gender (23, where 0 = female and 1 = male). The following shows parts of the regression output for a sample of 20 individuals. 21 Variable Coefficient 0.63 0.92 -0.51 S Sres = 112, SSexp = 84 Standard Error 0.09 0.19 0.92 23 (a) Compute the multiple coefficient of determination. (b) Interpret the estimated coefficient for x3.

Answer 1

a)

SSexplained = 84

SS residual = 112

SStotal = SSexplained + SS residual =84 + 112 = 196

Multiple co-efficient of determination = 1- SS residual / SStotal = 1- 112 / 196 = .4286

It means with the help of 3 regressor variable we are able to explain 42.86 % of variability in individual's income.

b)

If co-efficient of x3 is significant then it means when x1 and remain fixed then x3= 1 i.e a male tends to income 510 ( .51 *1000) dollars less than when x3=0 i.e female. In other words the income of a female is 510 more than a male with same age and level of education.

But if we do t test then it is seen t = -.52/ .92 = - 0.55

tcrit = t $\small \alpha$ /2 =0.025,18 = 2.101 > | t |

So reject the null hypothesis. Gender does not have effect on income. So males and females tend to income same when age & level of education are same.

c)

H0:  $\small \beta$2 = 0

H1:  $\small \beta$2 $\small \neq$ 0

$\small \hat{\beta }$= .92

SE($\small \hat{\beta }$) =.19

t = $\small \hat{\beta }$ / SE($\small \hat{\beta }$) = 4.84

tcrit = t $\small \alpha$ /2 =0.025,18 = 2.101 < | t |

So, we fail to reject null hypothesis.

So, level of education is significantly different from 0.

Feel free to comment if you need any clarification.

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