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Find the solution to these normal equations. The solutions are weighted least squares estimators of 0 and 1. 11-81. Consider a situation where both Y and X are random variables. Let sx and sy be the sample standard deviations of the observed x s and y s, respectively. Show that an alternative expression for the tted simple x is linear regression model y 0 1
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Is the estimator of the slope in the simple linear regression model unbiased 11-79. Suppose that we are tting a line and we wish to make the variance of the regression coef cient 1 as small as possible. Where should the observations xi, i 1, 2, p , n, be taken so as to minimize V( 1) Discuss the practical implications of this allocation of the xi. 11-80. Weighted Least Squares. Suppose that we , but the variance are tting the line Y 0 1x of Y depends on the level of x; that is, V1Yi 0 xi 2
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11-82. Suppose that we are interested in tting a simple linear regression model Y , 0 1x where the intercept, 0, is known. (a) Find the least squares estimator of 1. (b) What is the variance of the estimator of the slope in part (a) (c) Find an expression for a 100(1 )% con dence interval for the slope 1. Is this interval longer than the corresponding interval for the case where both the intercept and slope are unknown Justify your answer.
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IMPORTANT TERMS AND CONCEPTS In the E-book, click on any term or concept below to go to that subject. Analysis of variance test in regression Con dence interval on mean response Correlation coef cient Empirical models Con dence intervals on model parameters Least squares estimation of regression model parameters Model adequacy checking Prediction interval on a future observation Residual plots Residuals Scatter diagram Signi cance of regression Statistical tests on model parameters Transformations CD MATERIAL Lack of t test Logistic regression
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Historical Note Sir Francis Galton rst used the term regression analysis in a study of the heights of fathers (x) and sons ( y). Galton t a least squares line and used it to predict the son s height from the fathers height. He found that if a father s height was above average, the son s height would also be above average, but not by as much as the father s height was. A similar effect was observed for short heights. That is, the son s height regressed toward the average. Consequently, Galton referred to the least squares line as a regression line. Abuses of Regression. Regression is widely used and frequently misused; several common abuses of regression are brie y mentioned here. Care should be taken in selecting variables with which to construct regression equations and in determining the form of the model. It is possible to develop statistically signi cant relationships among variables that are completely unrelated in a causal sense. For example, we might attempt to relate the shear strength of spot welds with the number of empty parking spaces in the visitor parking lot. A straight line may even appear to provide a good t to the data, but the relationship is an unreasonable one on which to rely. You can t increase the weld strength by blocking off parking spaces. A strong observed association between variables does not necessarily imply that a causal relationship exists between those variables. This type of effect is encountered fairly often in retrospective data analysis, and even in observational studies. Designed experiments are the only way to determine causeand-effect relationships. Regression relationships are valid only for values of the regressor variable within the range of the original data. The linear relationship that we have tentatively assumed may be valid over the original range of x, but it may be unlikely to remain so as we extrapolate that is, if we use values of x beyond that range. In other words, as we move beyond the range of values of x for which data were collected, we become less certain about the validity of the assumed model. Regression models are not necessarily valid for extrapolation purposes. Now this does not mean don t ever extrapolate. There are many problem situations in science and engineering where extrapolation of a regression model is the only way to even approach the problem. However, there is a strong warning to be careful. A modest extrapolation may be perfectly all right in many cases, but a large extrapolation will almost never produce acceptable results.
Lack-of-Fit Test (CD Only)
Regression models are often t to data to provide an empirical model when the true relationship between the variables Y and x is unknown. Naturally, we would like to know whether the order of the model tentatively assumed is correct. This section describes a test for the validity of this assumption. The danger of using a regression model that is a poor approximation of the true functional relationship is illustrated in Fig. S11-1. Obviously, a polynomial of degree two or greater in x should have been used in this situation. We present a test for the goodness of t of the regression model. Speci cally, the hypotheses we wish to test are H0: The simple linear regression model is correct. H1: The simple linear regression model is not correct.