Introduction to Linear Regression Analysis. Douglas C. Montgomery

Introduction to Linear Regression Analysis - Douglas C. Montgomery


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detect this relationship has been obscured by the variance of the measurement process or that the range of values of x is inappropriate. A great deal of nonstatistical evidence and knowledge of the subject matter in the field is required to conclude that β1 = 0.

      In this section we consider confidence interval estimation of the regression model parameters. We also discuss interval estimation of the mean response E(y) for given values of x. The normality assumptions introduced in Section 2.3 continue to apply.

      2.4.1 Confidence Intervals on β0, β1, and σ2

      In addition to point estimates of β0, β1, and σ2, we may also obtain confidence interval estimates of these parameters. The width of these confidence intervals is a measure of the overall quality of the regression line. If the errors are normally and independently distributed, then the sampling distribution of both in29-1 and in29-2 is t with n − 2 degrees of freedom. Therefore, a 100(1 − α) percent confidence interval (CI) on the slope β1 is given by

      (2.39) image

      and a 100(1 − α) percent CI on the intercept β0 is

      (2.40) image

      These CIs have the usual frequentist interpretation. That is, if we were to take repeated samples of the same size at the same x levels and construct, for example, 95% CIs on the slope for each sample, then 95% of those intervals will contain the true value of β1.

      If the errors are normally and independently distributed, Appendix C.3 shows that the sampling distribution of (n − 2)MSRes/σ2 is chi square with n − 2 degrees of freedom. Thus,

ueqn29-1

      and consequently a 100(1 − α) percent CI on σ2 is

      Example 2.5 The Rocket Propellant Data

ueqn30-1

      or

ueqn30-2

      In other words, 95% of such intervals will include the true value of the slope.

      If we had chosen a different value for α, the width of the resulting CI would have been different. For example, the 90% CI on β1 is −42.16 ≤ β1 ≤ −32.14, which is narrower than the 95% CI. The 99% CI is −45.49 ≤ β1 ≤ 28.81, which is wider than the 95% CI. In general, the larger the confidence coefficient (1 − α) is, the wider the CI.

ueqn30-3

      From Table A.2, in30-3 and in30-4. Therefore, the desired CI becomes

ueqn30-4

      or

ueqn30-5

      (2.42) image

      To obtain a 100(1 − α) percent CI on E(y|x0), first note that in31-1 is a normally distributed random variable because it is a linear combination of the observations yi. The variance of in31-2 is

ueqn31-1

      since (as noted in Section 2.2.4) in31-3. Thus, the sampling distribution of

ueqn31-2

      is t with n − 2 degrees of freedom. Consequently, a 100(1 − α) percent CI on the mean response at the point x = x0 is

      Note that the width of the CI for E(y|x0) is a function of x0. The interval width is a minimum for in31-4 and widens as in31-5 increases. Intuitively this is reasonable, as we would expect our best estimates of y to be made at x values near the center of the data and the precision of estimation to deteriorate as we move to the boundary of the x


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