## Some Statistical Properties of Regression Results

There are some interesting results that go along with the analysis we have been developing For one thing, it can be shown [4] that the estimate of "a" is a random variable with Normal distribution. Such a conclusion should not come as a surprise since there is no particular reason why the distribution of values of "a" should favor the more pessimistic or the more optimistic value. All of the good attributes of the Normal distribution are working for us now: "a" is unbiased maximum likelihood estimator for the true value of "a" in the population. In turn, the expected value of "a" is the true value of "a" itself:

r2 = 1, then Y2distance/Y2dAv = 0, or ^distance = 0

Figure 8-5 shows the r2 for a data set.

x3 me ctangB in Cos* is explain&d by a ctiangs in T, fiowsver, some cost jwinls are no: explained by We linear progression ol T, such as Xl, X2. arm 5(3.

For this daia selh r* is iipprimmaUily Ă›.&5 meaning tbat 85% ol

Mean value of "a" Variance (a)

expected value of "a"

true value of "a" in the population s 2/ (X - Xav)2

Looking at the data observations of X for a moment, we write:

(Xi - Xav)2 = n * Variance (X) where n = number of observations of X in the population.

Now we have a couple of interesting results: as the sample size, n, gets larger, the variance of "a" gets smaller, meaning that we can zero in on the true value of "a" all that much better. Another way to look at this is that as the variance of X gets larger, meaning a larger spread of the X values in the population, again the variance of "a" is driven smaller, making estimate all the better. In effect, if all the X data are bunched together, then it is very difficult to find a line that predicts Y. We see that the ks in Figure 8-6 do not provide a sufficient spread to develop a regression curve.

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