www.downloadslide.com 11.6 Inferences Concerning Linear Functions of the Model Parameters: Simple Linear Regression 589 11.32 Refer to Exercises 11.5 and 11.17. a Is there sufficient evidence to indicate that the median sales price for new single-family houses increased over the period from 1972 through 1979 at the .01 level of significance? b Estimate the expected yearly increase in median sale price by constructing a 99% confidence interval. 11.33 Refer to Exercise 11.8 and 11.18. Is there evidence of a linear relationship between flow-through and static LC50s? Test at the .05 significance level. 11.34 Refer to Exercise 11.33. Is there evidence of a linear relationship between flow-through and static LC50s? a b Give bounds for the attained significance level. Applet Exercise What is the exact p-value? 11.6 Inferences Concerning Linear Functions of the Model Parameters: Simple Linear Regression In addition to making inferences about a single β i , we frequently are interested in making inferences about linear functions of the model parameters β 0 and β 1 .For example, we might wish to estimate E(Y ), given by E(Y ) = β 0 + β 1 x, where E(Y ) represents the mean yield of a chemical process for the settings of controlled process variable x or the mean mileage rating of four-cylinder gasoline engines with cylinder volume x. Properties of estimators of such linear functions are established in this section. Suppose that we wish to make an inference about the linear function θ = a 0 β 0 + a 1 β 1 , where a 0 and a 1 are constants (one of which may equal zero). Then, the same linear function of the parameter estimators, ˆθ = a 0 ˆβ 0 + a 1 ˆβ 1 , is an unbiased estimator of θ because, by Theorem 5.12, E(ˆθ) = a 0 E( ˆβ 0 ) + a 1 E( ˆβ 1 ) = a 0 β 0 + a 1 β 1 = θ. Applying the same theorem, we determine that the variance of ˆθ is V (ˆθ) = a 2 0 V ( ˆβ 0 ) + a 2 1 V ( ˆβ 1 ) + 2a 0 a 1 Cov( ˆβ 0 , ˆβ 1 ), where V ( ˆβ i ) = c ii σ 2 and Cov( ˆβ 0 , ˆβ 1 ) = c 01 σ 2 , with ∑ x 2 c 00 = i , c 11 = 1 , nS xx S xx c 01 = −x S xx .
www.downloadslide.com 590 Chapter 11 Linear Models and Estimation by Least Squares Some routine algebraic manipulations yield ⎛ ∑ ⎞ x 2 a 2 i 0 + a1 2 V (ˆθ) = ⎜ n − 2a 0a 1 x ⎟ ⎝ ⎠ σ 2 . S xx Finally, recalling that ˆβ 0 and ˆβ 1 are normally distributed in repeated sampling (Section 11.4), it is clear that ˆθ is a linear function of normally distributed random variables, implying that ˆθ is normally distributed. Thus, we conclude that Z = ˆθ − θ σˆθ has a standard normal distribution and could be employed to test the hypothesis H 0 : θ = θ 0 when θ 0 is some specified value of θ = a 0 β 0 + a 1 β 1 . Likewise, a 100(1 − α)% confidence interval for θ = a 0 β 0 + a 1 β 1 is ˆθ ± z α/2 σˆθ. We notice that, in both the Z statistic and the confidence interval above, σˆθ = √ V (ˆθ) is a constant (depending on the sample size n, the values of the x’s, and the values of the a’s) multiple of σ . If we substitute S for σ in the expression for Z, the resulting expression (which we identify as T ) possesses a Student’s t distribution in repeated sampling, with n − 2 df, and provides a test statistic to test hypotheses about θ = a 0 β 0 + a 1 β 1 . Appropriate tests are summarized as follows. A Test for θ = a 0 β 0 + a 1 β 1 H 0 : θ ⎧= θ 0 , ⎪⎨ θ>θ 0 , H a : θ t α , Rejection region: t < −t α , ⎩ |t| > t α/2 . Here, t α and t α/2 are based on n − 2 df.