Introduction to Probability and Statistics for Engineers and Scientists

370 Chapter 9: Regression

80 90 100 110 120 130

Deaths in 1988

120

110

100

90

80

Deaths in 1989

FIGURE 9.8 Scatter diagram of 1989 deaths versus 1988 deaths.

While the above supposition might be correct, it is important to realize that a regression
to the mean would probably have occurred even if none of the counties had done anything
out of the ordinary. Indeed, it could very well be the case that those counties having large
numbers of casualties in 1988 were just very unlucky in that year and thus a decrease
in the next year was just a return to a more normal result for them. (For an analogy, if
9 heads results when 10 fair coins are flipped then it is quite likely that another flip of these
10 coins will result in fewer than 9 heads.) Similarly, those counties having few deaths in
1988 might have been “lucky” that year and a more normal result in 1989 would thus lead
to an increase.
The mistaken belief that regression to the mean is due to some outside influence when
it is in reality just due to “chance” occurs frequently enough that it is often referred to as
theregression fallacy. ■

9.4.2 Inferences Concerningα

The determination of confidence intervals and hypothesis tests forαis accomplished in
exactly the same manner as was done forβ. Specifically, Proposition 9.3.1 can be used to
show that
√√
√
√

n(n−2)Sxx

∑

i

xi^2 SSR

(A−α)∼tn− 2 (9.4.3)

which leads to the following confidence interval estimator ofα.