Question

2. Using data from 2013 on 64 black females, the estimated linear regression between WAGE (earnings per hour, in S) and years of education, EDUC is WAGE8.451.99EDUC The standard error of the estimated slope coefficient is 0.52. Construct and interpret a 95% interval estimate for the effect of an additional year of education on a black female's expected hourly wage rate a. b. The standard error of the estimated intercept is 7.39. Test the null hypothesis that the intercept A-0 against the alternative that the true intercept is not zero, using the α-0.10 level of significance. In your answer, show (i) the formal null and alternative hypothesis, (i) the test statistic and its distribution under the null hypothesis,iii) the rejection region (in a figure of the t-distribution), (iv) the calculated value of the test statistic, and (v) state your conclusion, with its economic interpretation.

Answer 1

here the linear regaression y^= $\beta$ 1+ $\beta$ 2*X

or , $\hat{WAGE}$ =-8.45+1.99EDUC

here slope is $\beta$ 2=1.99 and intercept $\beta$ 1=-8.45

error df=n-k=64-2=62

n=number of observation=64 and

k=number of regression coefficient=2

(a) (1-alpha)*100% confidence interval for slope $\beta$ 2= $\hat{\beta}$ 2 ±t(alpha/2, error df)*SE( $\hat{\beta}$ )

95% confidence interval =1.99±t(0.05/2, 62)*0.52=1.99±2.00*0.52=1.99±1.04=(0.95,3.03)

(b) (i)here null hypothesis H0: $\beta$ 1=0 and

alternate hypothesis Ha: $\beta$ 1 $\ne$ 0

(ii) test statistic t=( $\hat{\beta}$ 1)/SE( $\hat{\beta}$ 1) will followin t-distribution with error df=n-k

(iii) rejection region |t|>t(0.05,62)=2.00

or, rejection region t<-2.00 or t>2.00

(iv) calculated t=-8.45/7.39=-1.1434

(v) since calcuted t=-1.1434 does not belongs rejection region, so we accept the null hypothesis H0 and conclude that intercept is zero. in other worlds no education no wage.

the resulting model implies that the response function must be exactly zero when all the predictors are set to zero or at their reference levels. For an ordinary regression model this means that the mean of the response variable is zero.