d.f. ( - 2). Note that the variance of 0 is (tVS„. Since is not known, we use an unbiased estimator RSS/(« - 2). Thus 6-/5„ is the estimated variance of and its square root is called the standard error denoted by SE(3).

We can follow a similar procedure for a. We substitute for in the variance of a and take the square root to get the standard error SE(a).

Thus we have the following result:

(a - a)/SE(a) and (0 - p)/SE(P) each have a /-distribution with d.f. ( - 2). These distributions can be used to get confidence intervals for a and and to test hypotheses about a and p.

d is usually called the standard error of the regression. It is denoted by SER (sometimes by SEE).

Illustrative Example

As an illustration of the calculation of standard errors, consider the data in Table 3.2. Eariier we obtained the regression equation of on x as

S = 3.6 + 0.75x

We will now calculate the standard errors of a and p. These are obtained by:

1. Computing the variances of a and p in terms of a.

2. Substituting 6-2 = RSS/(« - 2) for (r\

3. Taking the square root of the resulting expressions.

We have

V(a) F(P)

= a2

I X

\n 5, - g - 0.036,.

(21)2 28

= 1.83

- 2

SE(a) = \/(2.39)(1.83) = 2.09 SE(P) = V(0.036)(1.83) = 0.256

The standard errors are usually presented in parentheses under the estimates of the regression coefficients.

Confidence Intervals for a, p, and

Since (a - a)/SE(a) and (0 - P)/SE(0) have /-distributions with d.f. (« - 2), using the table of the /-distribution (in the appendix) with « - 2 = 8 d.f. we get

Prob

Prob

-2.306 < < 2.306

SE(a)

= 0.95

* 0.95

These give confidence intervals for a and p. Substituting the values of ct, , SE(a), and SE(P) and simplifying we get the 95% confidence limits for a and as (-1.22,8.42) for a and (0.16,1.34) for p. Note that the 95% confidence limits f(M- a are a a: 2.306 SE(a) and for p are p ± 2.306SE(P). Although it is not often used, we will also illustrate the pitjcedure of finding a confidence ittterval fc«r a. Since we know that RSS/ff has a x-distribution with d.f. (n -2), we can use the tables of the x-distribution for getting any required confidence interval. Suppose that we want a 95% confidence interval. Note that we will write RSS = ( - 2)6.

From the tables of the x-distribution with degrees of freedom 8, we find that the probability of ctaining a value < 2.18 is 0.025 and of obtaining a value > 17.53 is 0.025. Hence

Prob 2.18 < < 17.53 = 0.95

Since = 1.83, substituting this value, we get the 95% confidence limits for 0(0.84, 6.72).

Note that the confidence intervals for a and p are symmetric around a and p, respectively (because the /-distribution is a symmetric distribution), "niis is not the case with the confidence interval for o. It is not symmetric around CT.

The confidence intervs we have obtained for a, p, and cr are all very wide. We can produce narrower intervals by reducing the confidence coefficient. For instance, the 80% confideiKie limits for p are

P ± 1.397SE(p) since Prob<-1.397 < t < 1.397) = 0.80

from the Mables with 8 d.f. We therefore get the 80% confidence limits for as

0.75 ± 1.397(0.256) = (0.39, 1.11)

The confidence intervals we have constructed are two-sided intervals. Sometimes we want upper or lower limits for p. In which case we construct onesided intervals. For instaiK:e, firom the f-tables with 8 d.f. we have

ProbCl < 1.86) = 0.95

Testing of Hypotheses

Turning to the problem of testing of hypotheses, suppose we want to test the hypothesis that the true value of p is 1.0. We know what

p - P has a /-distribution with--~ SE(p) (n - 2) degrees of freedom

Let / be the observed /-value. If the alternative hypothesis is p 1 then we consider /j as the test statistic. Hence if the true value of p is 1.0, we have

0 = "0 26 " "" "

Looking at the /-tables for 8 degrees of freedom, we see that

Prob(/ > 0.706) = 0.25

Prob(/ > 1.397) = 0.10

Thus Prob(/ > 0.98) is roughly 0.19 (by linear interpolation) or Prob(/ > 0.98) - 0.38. This probability is not very low and we do not reject the hypothesis that p = 1.0. It is customary to use 0.05 as a low probability and to reject the suggested hypothesis if the probability of obtaining as extreme a /-value as the observed /q is less than 0.05. In this case either the suggested hypothesis is not true or it is true but an improbable event has occurred.

Note that for 8 degrees of freedom, the 5% probabiUty points are ± 2.306 for a two-sided test and ±1.86 for a one-sided test. Thus if both high and low /-values are to be considered as evidence against the suggested hypothesis, we reject it if the observed / is > 2.306 or < -2.306. On the other hand, if only very high or very low /-values are to be considered as evidence against the suggested hypothesis, we reject it if /„ > 1.86 or /„ < - 1.86, depending on the suggested direction of deviation.

Prob(r> -1.86) = 0.95

Hence for a one-sided confidence interval, the upper limit for p is

P + 1.86SE(p) = 0.75 + 1.86(0.256) = 0.75 + 0.48 = 1.23

The 95% confidence interval is (-oo, 1.23). Similarly, the lower limit for p is

p - 1.86SE(P) = 0.75 - 0.48 = 0.27

Thus the 95% confidence interval is (0.27, + oo).

We will give further examples of the use of one-sided intervals after we discuss tests of hypotheses.