(Continuation from Chapter 4)Sir Francis Galton, a Cousin of James $\widehat {\text { Studenth }}$

Essay

(Continuation from Chapter 4)Sir Francis Galton, a cousin of James Darwin, examined the relationship between the height of children and their parents towards the end of the 19^th century. It is from this study that the name "regression" originated. You decide to update his findings by collecting data from 110 college students, and estimate the following relationship: $\widehat {\text { Studenth }}$ = 19.6 + 0.73 × Midparh, R² = 0.45, SER = 2.0
(7.2)(0.10)
where Studenth is the height of students in inches, and Midparh is the average of the parental heights. Values in parentheses are heteroskedasticity robust standard errors. (Following Galton's methodology, both variables were adjusted so that the average female height was equal to the average male height.)
(a)Test for the statistical significance of the slope coefficient.
(b)If children, on average, were expected to be of the same height as their parents, then this would imply two hypotheses, one for the slope and one for the intercept.
(i)What should the null hypothesis be for the intercept? Calculate the relevant t-statistic and carry out the hypothesis test at the 1% level.
(ii)What should the null hypothesis be for the slope? Calculate the relevant t-statistic and carry out the hypothesis test at the 5% level.
(c)Can you reject the null hypothesis that the regression R² is zero?
(d)Construct a 95% confidence interval for a one inch increase in the average of parental height.