Suppose we have a random sample of 50 people and their weight, W and height, H are recorded to the nearest pound and inch respectively. A regression of W on H and an intercept gives:
Wi = 99.41(6.45) + 3.94(1.86)Hi
R^2 = 0.81, SER = \(\sqrt{Su^2}\) = 10.1
(Standard errors are given in parentheses)
(a). What is the prediction of the weight of someone who is 50inches tall, 60inches tall, 70 inches tall?
(b). What is the prediction for the change in weight of someone who grows 1.5 inches? What is the predicted difference in weight between 2 people who differ by 6 inches in height?
(c). Suppose that the sample average of H is 56 inches. What is the sample average of W? Why?
(d). Perform 2-sided t-tests for the null hypothesis that the slope is zero at the 5% and 1% significance levels and interpret the results.
(e). Perform 1-sided t-tests for the null hypothesis that the slope is zero at the 5% and 1% significance levels and interpret the results. Ensure you justify the use of a 1-sided test.
(f). Explain the potential advantage in relation to the sizes of Type I and Type II errors of using a 1-sided t test rather than a 2-sided t-test if a one-sided test is justified. [Diagrams are likely to be helpful.]
(g). Suppose we measured weight in kilos. What will the new intercept be?
(h). Prove that if Wi = \(\lambda1 + \lambda2Wi\lambda\) \(\lambda1 + \lambda2b1\) where b1 is the intercept of the original regression of W on H.
(i). Explain how the standard error of the slope coefficient in () is related to the standard error of the slope in the original regression.
(j.) Suppose that in addition to measuring weight in kilos we also measured height in centimetres. What would the new slope estimate be?
(k). Prove that if W1 = \(\lambda1 + \lambda2Wi\) and Hi = \(\mu1 + \mu2Hi\) then the new slope is b2 = \((\lambda2/\mu2)b2\)
where b2 is the slope of the original regression of W on H.
(l). What will R^2 be in the regression you considered in (j)? Why?
(m). Explain how the OLS estimator of the variance of the disturbance term in (k) is related to that in (h).
(n). Explain how the standard error of the slope coefficient in (k) is related to that in (h).