the ed. 11.2 General Concepts 459 TABLE 11.1 Sample data from the Greene-Touchstone study relating birthweight and estriol level in pregnant women near term Estriol Birthweight (mg/24 hr) (g/100) Estriol (mg/24 hr) Birthweight (g/100) X; Y₁ Xi Yi 2 12 7 25 17 17 32 9 25 18 25 32 3 9 25 19 27 34 4 12 27 20 15 34 5 14 27 21 15 34 6 16 27 22 15 35 7 16 24 23 16 35 8 14 30 24 19 34 9 16 30 25 18 35 10 16 31 26 17 36 11 17 30 27 18 37 12 19 31 28 20 38 13 21 30 29 22 40 14 24 28 30 25 39 15 15 32 31 24 43 16 16 32 EQUATION 11.2 DEFINITION 11.2 EXAMPLE 11.4 EXAMPLE 11.5 Source: Based on the American Journal of Obstetrics and Gynecology, 85(1), 1-9, 1963. Let's assume e follows a normal distribution, with mean 0 and variance o². The full linear-regression model then takes the following form: y = a+ẞx+e where e is normally distributed with mean 0 and variance σ². For any linear-regression equation of the form y=a+ẞx+e, y is called the depen- dent variable and x is called the independent variable because we are trying to predict y as a function of x. Obstetrics Birthweight is the dependent variable and estriol is the independent variable for the problem posed in Example 11.3 because estriol levels are being used to try to predict birthweight. One interpretation of the regression line is that for a woman with estriol level x, the corresponding birthweight will be normally distributed with mean a + ẞx and variance o². If σ2 were 0, then every point would fall exactly on the regression line, whereas the larger o² is, the more scatter occurs about the regression line. This rela- tionship is illustrated in Figure 11.2. How can ẞ be interpreted? If ẞ is greater than 0, then as x increases, the expected value of y = a + ẞx will increase. Obstetrics This situation appears to be the case in Figure 11.3a for birthweight (y) and estriol (x) because as estriol increases, the average birthweight correspondingly increases. If ẞ is less than 0, then as x increases, the expected value of y will decrease.

the ed. 11.2 General Concepts 459 TABLE 11.1 Sample data from the Greene-Touchstone study relating birthweight and estriol level in pregnant women near term Estriol Birthweight (mg/24 hr) (g/100) Estriol (mg/24 hr) Birthweight (g/100) X; Y₁ Xi Yi 2 12 7 25 17 17 32 9 25 18 25 32 3 9 25 19 27 34 4 12 27 20 15 34 5 14 27 21 15 34 6 16 27 22 15 35 7 16 24 23 16 35 8 14 30 24 19 34 9 16 30 25 18 35 10 16 31 26 17 36 11 17 30 27 18 37 12 19 31 28 20 38 13 21 30 29 22 40 14 24 28 30 25 39 15 15 32 31 24 43 16 16 32 EQUATION 11.2 DEFINITION 11.2 EXAMPLE 11.4 EXAMPLE 11.5 Source: Based on the American Journal of Obstetrics and Gynecology, 85(1), 1-9, 1963. Let's assume e follows a normal distribution, with mean 0 and variance o². The full linear-regression model then takes the following form: y = a+ẞx+e where e is normally distributed with mean 0 and variance σ². For any linear-regression equation of the form y=a+ẞx+e, y is called the depen- dent variable and x is called the independent variable because we are trying to predict y as a function of x. Obstetrics Birthweight is the dependent variable and estriol is the independent variable for the problem posed in Example 11.3 because estriol levels are being used to try to predict birthweight. One interpretation of the regression line is that for a woman with estriol level x, the corresponding birthweight will be normally distributed with mean a + ẞx and variance o². If σ2 were 0, then every point would fall exactly on the regression line, whereas the larger o² is, the more scatter occurs about the regression line. This rela- tionship is illustrated in Figure 11.2. How can ẞ be interpreted? If ẞ is greater than 0, then as x increases, the expected value of y = a + ẞx will increase. Obstetrics This situation appears to be the case in Figure 11.3a for birthweight (y) and estriol (x) because as estriol increases, the average birthweight correspondingly increases. If ẞ is less than 0, then as x increases, the expected value of y will decrease.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.3: Least Squares Approximation

Problem 31EQ

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