Description of the Test.

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LEAST-SQUARES REGRESSION

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First we describe how the least-squares test of

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{3 = 0 is performed. Justification of the procedure is given afterward. To test {3 = 0, first use (3.2) to calculate ~, and then calculate

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I -n - 2

(3.3)

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Then calculate an estimate of the standard deviation of ~ by substituting for (J' in the formula

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SD( {3}

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(3.4)

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where (J' denotes the standard deviation of the population of errors. The test statistic is

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II I

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est.SD(~)

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(3.5)

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(We use absolute value notation for III because later we talk about I = ~ / est.SD(~).) The p-value of the test is obtained from the I distribution with n - 2 degrees of freedom. The p-value is calculated as the probability that the absolute value of a random variable with this distribution is greater than or equal to the value of II I calculated from (3.5).

Justification of the Test. The most relevant information we have about {3 is the estimate ~. The value of ~ should indicate whether {3 = 0 or not. There is strong evidence that {3 =1= 0 when ~ is "far" from O. For the acid content data we calculated ~ = 0.3216. Is this far from O It depends on how variable ~ is. When we speak of the variability of ~, we are thinking of ~ as a random variable. Imagine repeating the acid content experiment an infinite number of times. The infinite number of values of ~ obtained in these experiments would be centered around {3 but they would vary. The size of a typical deviation of ~ from {3 is measured by the standard deviation of ~. We can estimate SD(~) by substituting (3.3) into (3.4) to obtain est.SD(~). For the acid content data, est.SD(~) = 0.0056. So the distance between ~ = 0.3216 and 0 is about 57 (= 0.3216/0.0056) times the size of SD(~). This makes it very unlikely that {3 = o. Thus we see that a reasonable test of {3 = 0 can be based on the test statistic II I in (3.5). A very large value of II I means that ~ is much farther

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{3 = 0

from 0 than would be expected if {3 = against {3 = O.

This constitutes strong evidence

Reasonableness of the Formula for SD(P). Consider formula (3.4). It implies that SD(~) is smaller when (7' is smaller, that is, when the variability of the random errors is smaller. This makes sense since we should be able to estimate {3 more accurately in the presence of smaller errors. Also, SD(~) is smaller when L(x i - i)2 is larger, that is, when the x-values are more spread out. This also makes sense if we think about the following analogy. Suppose we want to use a pencil and ruler to draw a horizontal line 10 centimeters above the bottom of a sheet of paper. We could draw two points 10 centimeters above the bottom and then align the ruler on these two points and draw a line. Because of human error, the line would not be exactly horizontal. To get the line to be as horizontal as possible, we would choose the two points to be on opposite edges of the sheet of paper, that is, as spread out as possible. Estimating (J". A natural estimate of the standard deviation of the population of errors is the standard deviation of the sample of estimated errors, that is, of the residuals i = Yi ~Xi. This estimate is I( n - 1) , but instead of using this, we modify it slightly and use formula (3.3). The modification has the nice feature that &2 is an unbiased estimate of (7'2. The divisor n - 2 in formula (3.3) is sometimes called the degrees of freedom of the estimate &2. The subtraction of 2 from n corresponds to the fact that we must estimate two parameters a and {3 in order to form the residuals i