Inferences Concerning m Y x and

12.4 Inferences Concerning m Y x and

  the Prediction of Future Y Values

  Let x denote a specified value of the independent variable x. Once the estimates

  bˆ 0 and bˆ 1 have been calculated, bˆ 0 1 bˆ 1 x can be regarded either as a point esti- mate of m Y x (the expected or true average value of Y when x 5 x ) or as a prediction of the Y value that will result from a single observation made when

  CHAPTER 12 Simple Linear Regression and Correlation

  x 5 x . The point estimate or prediction by itself gives no information concerning

  how precisely m Y x has been estimated or Y has been predicted. This can be reme-

  died by developing a CI for m Y x and a prediction interval (PI) for a single Y value. Before we obtain sample data, both bˆ 0 and bˆ 1 are subject to sampling

  variability—that is, they are both statistics whose values will vary from sample to

  sample. Suppose, for example, that b 0 5 50 and b 1 52 . Then a first sample of (x, y) pairs might give bˆ 0 5 52.35, bˆ 1 5 1.895 ; a second sample might result in bˆ 0 5 46.52, bˆ 1 5 2.056 ; and so on. It follows that Yˆ 5 bˆ 0 1 bˆ 1 x itself varies in

  value from sample to sample, so it is a statistic. If the intercept and slope of the population line are the aforementioned values 50 and 2, respectively, and x 5 10, then this statistic is trying to estimate the value

  50 1 2(10) 5 70 . The

  estimate from a first sample might be

  52.35 1 1.895(10) 5 71.30 , from a second

  sample might be

  46.52 1 2.056(10) 5 67.08 , and so on.

  This variation in the value of bˆ 0 1 bˆ 1 x can be visualized by returning to

  Figure 12.13 on page 492. Consider the value x 5 300 . The heights of the 20 pic- tured estimated regression lines above this value are all somewhat different from one another. The same is true of the heights of the lines above the value x 5 350 . In

  fact, there appears to be more variation in the value of bˆ 0 1 bˆ 1 (350) than in the value

  of bˆ 0 1 bˆ 1 (300) . We shall see shortly that this is because 350 is further from x 5 235.71 (the “center of the data”) than is 300.

  Methods for making inferences about b 1 were based on properties of the sampling distribution of the statistic bˆ 1 . In the same way, inferences about the

  mean Y value b 0 1b 1 x are based on properties of the sampling distribution of

  the statistic bˆ 0 1 bˆ 1 x . Substitution of the expressions for bˆ 0 and bˆ 1 into bˆ 0 1 bˆ 1 x followed by some algebraic manipulation leads to the representation of bˆ 0 1 bˆ 1 x as a

  linear function of the Y i ’s:

  2 x) 2 dY i 5 d i Y i51 i i i51

  n

  1 (x 2 x)(x

  2 x)

  bˆ 0 1 bˆ 1 x 5 g 1 c i g

  The coefficients d 1 ,d 2 , c, d n in this linear function involve the x i ’s and x, all of

  which are fixed. Application of the rules of Section 5.5 to this linear function gives the following properties.

  PROPOSITION

  Let , Yˆ 5 bˆ 0 1 bˆ 1 x where x is some fixed value of x. Then

  1. The mean value of Yˆ is E(Yˆ) 5 E(bˆ 0 1 bˆ 1 x) 5 m bˆ 0 1bˆ 1 x 5b 0 1b 1 x

  Thus bˆ 0 1 bˆ 1 x is an unbiased estimator for b 0 1b 1 x (i.e., for m Y x ).

  2. The variance of is Yˆ

  V(Yˆ) 5 s

  and the standard deviation s Yˆ is the square root of this expression. The

  estimated standard deviation of bˆ 0 1 bˆ 1 x , denoted by s Yˆ or s bˆ 0 1bˆ 1 x , results

  from replacing s by its estimate s:

  12.4 Inferences Concerning m Y • x and the Prediction of Future Y Values

  3. Yˆ has a normal distribution.

  The variance of bˆ 0 1 bˆ 1 x is smallest when x 5 x and increases as x moves away from in either direction. Thus the estimator of m Y x is more precise when x is near

  the center of the x i ’s than when it is far from the x values at which observations have been made. This will imply that both the CI and PI are narrower for an x near than

  for an x far from . Most statistical computer packages will provide both bˆ 0 1 bˆ 1 x

  and s bˆ 0 1bˆ 1 x for any specified x upon request.