Question

One field of economics, referred to as human capital, has often held that a person’s income (I) could be determined on the basis of his or her (1) education level (E), (2) training (T), and (3) the person’s general level of health (H). Using 25 employees at a small textile firm in North Carolina, a researcher regressed income on the other three variables and got the following results.

I = 27.2 + 3.7 E + 1.7 T + 3.05 H

(3.70) (6.21) (4.32) (6.79)

with r-squared = 0.67 and the calculated F statistic equal to 5.97. Beneath each coefficient is the calculated t statistic.

I is measured in units of $1000, E and T are measured in years, and H is measured in terms of a scaled index of one’s health: the higher the index, the better the level of health.

a. If one’s education increases by two years, what happens to his or her income?

b. Is the model significant at the 5% level? Show all parts of any test.

c. Determine which variable(s) is (are) significant at the 10 percent level. Show all parts of any test.

d. What is the value of the adjusted coefficient of determination?

Answer 1

Answer:

Given Data

Part A:

The regression equation is given as,

I = 27.2 +3.7 E +1.7 T +3.05 * H

Where,

I denotes a person’s income.

E denotes his or her education level.

T denotes his or her training.

H denotes the person’s general level of health.

If E is increased by 2 years,

Then, 11 = 27.2+3.7 E+2)+1.7*T+3.05 H = 34.6+3.7*E+1.7*T+3.05* H

So, (11 - 1) = 7.4

Hence, Income is supposed to increase by 7.4 units(in $1000) in case his/her education increases by 2 years.

Part B:

It is given that the F-statistic, F0 = 5.97.

We know that the sample size, n = 25, and there are k= 3 independent variable.

For testing the significance of regression, We have

null hypothesis, HO: B1 = 32 = 83 = 0

and alternate hypothesis, H: B; +0 for at least one j € {1,2,3}

Where are coefficient of regression.

Now we know that F-statistic will follow an or F3,21 distribution.

We reject H0 at 5% level of significance if Fo > F 0.05,3,21 .

Using R, we get that F 0 .05,3,21 = 3.07 [ R code: qf(0.95,3,21) ]

So F0 = 5.97 > 3.07 = critical value.

So we reject H0 at 5% level of significance and conclude that the model is significant.

Part C:

For significance of a variable, we have to test the

null hypothesis, Ho : B; = 0,3 € {1,2,3}

and alternate hypothesis, H1: B; +0

Where are coefficient of regression.

This test leads to a t-test.

We have the t-statistic as t1 = 6.21, t2 = 4.32, t3 = 6.79 .

We reject H0 at 10% level of significance if t - statistic > or 0.05,21

Using R we get +0.05,21 = 1.72 . [ R code: qt(0.95,21) ]

Here t; > 1.72 = t0.05,21 for all j = 1,2,3 .

So we can say that all the variables are significant at 10% level of significance.

Part D:

The adjusted coefficient of determination is,

(1 - R)(n-1) (n-k – 1)

We have R2 = 0.67, n = 25, k = 3.

Thus R =1- (1 -0.67) (25 - 1) 2 = 0.62 (approx) (25 – 3-1) .

Hence value of adjusted coefficient of determination is 0.62.

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