shear strength and age were perfectly deterministic (no er-
ror). Does this plot indicate that age is a reasonable choice
of regressor variable in this model?
11-13. Show that in a simple linear regression model
the point ( ) lies exactly on the least squares regression
line.
11-14. Consider the simple linear regression model Y 0
1 x. Suppose that the analyst wants to use zx as the
regressor variable.
(a) Using the data in Exercise 11-12, construct one scatter
plot of the ( ) points and then another of the
( ) points. Use the two plots to intuitively
explain how the two models, Y 0 1 xand
, are related.
(b) Find the least squares estimates of and in the model. How do they relate to the least
squares estimates ˆ 0 and ?ˆ 1
Y* 0 * 1 z* 0 * 1Y* 0 * 1 zzixi x, yixi, yixx, y11-15. Suppose we wish to fit the model
, where (i 1, 2, p , n). Find
the least squares estimates of and. How do they relate
to and?
11-16. Suppose we wish to fit a regression model for which
the true regression line passes through the point (0, 0). The ap-
propriate model is Yx. Assume that we have npairs
of data (x 1 , y 1 ), (x 2 , y 2 ),p, (xn, yn). Find the least squares esti-
mate of .
11-17. Using the results of Exercise 11-16, fit the model
Yxto the chloride concentration-roadway area
data in Exercise 11-11. Plot the fitted model on a scatter
diagram of the data and comment on the appropriateness of
the model.ˆ 0 ˆ 1* 0 * 1* 11 xi x 2 i y*iyi yy*i* 011-3 PROPERTIES OF THE LEAST SQUARES ESTIMATORS 383Observation Strength y Age x
Number (psi) (weeks)
1 2158.70 15.50
2 1678.15 23.75
3 2316.00 8.00
4 2061.30 17.00
5 2207.50 5.00
6 1708.30 19.00
7 1784.70 24.00
8 2575.00 2.50
9 2357.90 7.50
10 2277.70 11.00Observation Strength y Age x
Number (psi) (weeks)
11 2165.20 13.00
12 2399.55 3.75
13 1779.80 25.00
14 2336.75 9.75
15 1765.30 22.00
16 2053.50 18.00
17 2414.40 6.00
18 2200.50 12.50
19 2654.20 2.00
20 1753.70 21.5011-3 PROPERTIES OF THE LEAST SQUARES ESTIMATORSThe statistical properties of the least squares estimators and may be easily described.
Recall that we have assumed that the error term in the model Y 0 1 xis a random
variable with mean zero and variance ^2. Since the values of xare fixed, Yis a random vari-
able with mean 0 1 xand variance ^2. Therefore, the values of and depend
on the observed y’s; thus, the least squares estimators of the regression coefficients may be
viewed as random variables. We will investigate the bias and variance properties of the least
squares estimators and.
Consider first. Because is a linear combination of the observations Yi, we can use
properties of expectation to show that expected value of is(11-15)Thus, is an ˆ 1 unbiased estimatorof the true slope 1.E 1 ˆ 12 1ˆ 1ˆ 1 ˆ 1ˆ 0 ˆ 1Y (^0) x ˆ 0 ˆ 1
ˆ 0 ˆ 1
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