We have defined the simple linear regression model to be y b1 b2x e. Suppose however that we knew, for a fact,

We have defined the simple linear regression model to be y ¼ b1 þ b2x þ

e. Suppose however that we knew, for a fact, that b1 ¼ 0.

(a) What does the linear regression model look like, algebraically, if b1 ¼ 0?

(b) What does the linear regression model look like, graphically, if b1 ¼ 0?

(c) If b1 ¼ 0 the least squares ‘‘sum of squares’’ function becomes Sðb2Þ ¼

N i¼1ðyi b2xiÞ

2

. Using the data,

plot the value of the sum of squares function for enough values of b2 for you to locate the approximate minimum. What is the significance of the value of b2 that minimizes Sðb2Þ? (Hint: Your computations will be simplified if you algebraically expand Sðb2Þ ¼ N i¼1ðyi b2xiÞ
2 by squaring the term in parentheses and carrying the summation operator through.)
(d)^Using calculus, show that the formula for the least squares estimate of b2 in this model is b2 ¼ xiyi= x2 i . Use this result to compute b2 and compare this value to the value you obtained geometrically.

(e) Using the estimate obtained with the formula in (d), plot the fitted (estimated)
regression function. On the graph locate the point ðx; yÞ. What do you observe?

(f) Using the estimates obtained with the formula in (d), obtain the least squares residuals, ^ei ¼ yi b2xi. Find their sum.
(g) Calculate xi^ei.

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