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We argued earlier in this section that a very large value of {3 = 0 because such a value of It I would be very unlikely if {3 = O. In other words, we can measure the strength of the observed data's evidence against {3 = 0 by how unlikely the observed It I would be if {3 = o. This is the idea behind p-values. To define the p-value of the test, we must distinguish between two different views of It I in (3.5). The first view is to view It I as the observed value of It I, that is, the value obtained by substituting the observed values of the y/s into the formula. The second view is to view the y/s and hence It I as random variables. The p-ualue of the test is the probability, assuming {3 = 0, that the random variable It I is as large or larger than the observed It I. To interpret this, suppose model (3.1) is true with {3 = 0 and imagine the acid content experiment is repeated an infinite number of times. For each experiment, the test statistic It I would be calculated. The p-value is the
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proportion of this imagined infinite collection of It I's that would be as large or larger than the value of It I calculated from the actual experiment. If the p-value is very small, then it is very unlikely that {3 = 0 and so we conclude {3 "" O. Otherwise, we conclude {3 = O. It can be left to the judgement of the individual to say when a p-value is "very small". Most statisticians would consider 0.01 to be very small and not many would consider 0.10 to be very small. For p-values between 0.01 and 0.10 judgements vary. Note the difference in our confidence about the two conclusions {3 "" 0 and {3 = O. When the p-value is very small and we conclude {3 "" 0, we are quite sure about our conclusion because the evidence is strong. But when we conclude {3 = 0, we are not convinced that {3 = 0 exactly but are only concluding that {3 = 0 is plausible and provides an adequate model for the data. The Distribution of t. In order to calculate the p-value we must know the probability distribution of the random variable It I, or of t, when {3 = O. The test statistic t is obtained from the data, that is, from the x /s and y/s. The x /s are assumed to be nonrandom constants, so the randomness of t derives from the randomness of the y/s. Note that the probability distribution of Yi is not completely known, even if we suppose {3 = 0, because then Yi = a + e i and a is an unknown parameter and the distribution of e i is not completely known; in particular, the standard deviation u of ei cannot be assumed to be known. However, by inspecting the formula for t it can be seen that t depends on the y/s only through the difference Yi - y, and the parameter a cancels in these differences. Moreover, the parameter u cancels in the ratio of ~ to est.SD(~}. Therefore, if we specify the shape of the distribution of the random errors, leaving only the standard deviation unknown, then the distribution of t is completely known. If we specify the shape to be normal (bell-shaped), that is, if we assume the normal linear regression model, then the resulting distribution of t, when {3 = 0, is called the t distribution with n - 2 degrees of freedom. (The degrees of freedom are associated with the estimate a of u.) Even if the distribution of the random errors is not normal, the distribution of the test statistic t, when {3 = 0, is still close to the t distribution with n - 2 degrees of freedom, provided that the sample size n is large. It should be safe to rely on the t distribution with a sample size as large as 20, provided the normality of the distribution of the random errors has been checked (see the next section) and the data, if necessary, have been transformed. The Acid Content Data. For the acid content data, the degrees of freedom are 18 (= 20 - 2). Above we calculated It I to be about 57. We can use the t table in the Appendix to pin down the p-value. Looking in the table
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