Question

We have 2008 data on y = income per capita (in thousands of dollars) and x = percentage of the population with a bachelor’s degree or more for the 50 U.S. states plus the District of Columbia, a total of N = 51 observations. We have results from a simple linear regression of y on x. a. The estimated error variance is ?̂ 2 = 14.24. What is the sum of squared least squares residuals? b. The estimated variance of ?2 is 0.0092. What is the standard error of ?2? What is the value of ∑(?? − ?̅) 2 ? c. The estimated slope is ?2 = 1.0290. Interpret this result. d. Using ?̅= 27.3569 and ?̅ = 39.6689, calculate the estimate of the intercept. e. Given the results in (b) and (d), what is ∑?? 2 ? f. For the state of Georgia, the value of ? = 34.893 and ? = 27.5. Compute the least squares residual, using the information in parts (c) and (d).

Answer 1

(a) The error variance is (since there are two estimates including the intercept) and hence, or .

(b) The standard error of b2 would be or or .

The variance slope would be as , and hence, or or .

(c) The slope is the change in y on average for a unit change in x. For b2 be 1.0290, we may say that for a unit increase in percentage of population with a bachelor's degree, the income per capita would increase by 1.0290 thousand of dollars, on average.

(d) The estimate of the intercept would be or or .

(e) We have the sum of square of deviation of x's as , and for the given mean of x, we have

or

or

or .

Now, since , we have . Hence, we have

or or .

(f) We have or or or or .