Introduction to Linear Regression Analysis. Douglas C. Montgomery

Introduction to Linear Regression Analysis - Douglas C. Montgomery


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      where we have specified a two-sided alternative. Since the errors εi are NID(0, σ2), the observations yi are NID(β0 + β1xi, σ2). Now in23-1 is a linear combination of the observations, so in23-2 is normally distributed with mean β1 and variance σ2/Sxx using the mean and variance of in23-3 found in Section 2.2.2. Therefore, the statistic

ueqn23-1

      (2.25) image

      Alternatively, a P-value approach could also be used for decision making.

      The denominator of the test statistic, t0, in Eq. (2.24) is often called the estimated standard error, or more simply, the standard error of the slope. That is,

      (2.26) image

      Therefore, we often see t0 written as

      A similar procedure can be used to test hypotheses about the intercept. To test

      (2.28) image

      we would use the test statistic

      (2.29) image

      A very important special case of the hypotheses in Eq. (2.23) is

      (2.30) image

image image

      The test procedure for H0: β1 = 0 may be developed from two approaches. The first approach simply makes use of the t statistic in Eq. (2.27) with β10 = 0, or

ueqn25-1

      The null hypothesis of significance of regression would be rejected if |t0| > tα/2,n−2.

      Example 2.3 The Rocket Propellant Data

      We test for significance of regression in the rocket propellant regression model of Example


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