Question

Suppose that (Yi, Xi) satisfy the least squares assumptions in Key Concept 4.3 and, in addition, ui is N(0, σ2 u) and is independent of Xi. A sample of size n = 30 yields = 43.2 + 61.5X, R2 = 0.54, SER = 1.52, (10.2) (7.4) where the numbers in parentheses are the homoskedastic-only standard errors for the regression coefficients.

a) Construct a 95% confidence interval for β0.

b) Test H0: β1 = 55 vs. H1 : β1 ≠ 55 at the 5% level.

c) Test H0: β1 = 55 vs. H1 : β1 > 55 at the 5% level.

Answer 1

Ans. Standard error of b0, s0 = 10.2

Standard error of b1, s1 = 7.4

t critical at 95% level of confidence and 29 degrees of freedom, t = 2.04523

a) 95% confidence interval of b0 = [b0 - s0*z , b0 + s0*z] = [43.2 - 7.4*2.04523, 43.2 + 7.4*2.04523] = [28.0653, 58.3347]

b) t statistic = (b1 - Hypothesised B1)/s1 = (61.5 - 55)/7.4 = 0.87837

As tstatistic < t critical, so, we fail to reject the null hypothesis at 5% level of significance.

C) t statistic = (b1 - Hypothesised B1)/s1 = (61.5 - 55)/7.4 = 0.87837

This hypotheses has alternate hypothesis as b1>55, so, it is an upper tailed test,

Thus, upper tailed t critical = 1.699127

As t statistic < t critical, so, we fail to reject the null hypothesis at 5% level of significance.

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