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11-76. Consider the simple linear regression model 2 Y , with E( ) 0, V( ) , and the 0 1x errors uncorrelated. (a) Show that cov 1 0, 1 2 x 2 Sx x. 2 (b) Show that cov 1Y, 1 0. 11-77. Consider the simple linear regression model 2 Y , with E( ) 0, V( ) , and the 0 1x errors uncorrelated. 2 (a) Show that E( 2 ) E(MSE) . 2 2 (b) Show that E(MSR) 1 Sx x. 11-78. Suppose that we have assumed the straight-line regression model Y

0 1x1

Below, we follow the convention that upper case letters represent 32bit words, while lower case represent 8-bit bytes- for the one require 16-bit quantity, we also use lower case. All arithmetic is to be taken modulo 232. As in other sections of this book, we adopt the convention that bits are numbered from left-to-right, beginning with 0. In the PKZIP attack discussed below, we often need t o specify a range of bits within a byte or a 32-bit word. We use the notation (A)i,..j, where j 2 i , for the string of bits of length j - i 1 beginning with bit i and ending with bit j of A. The PKZIP encryption algorithm appears in Table 3.11. Here, p is the current plaintext byte, c the resulting ciphertext byte, and k is the keystream byte.

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where the wi are constants, often called weights. Show that for an objective function in whole each squared residual is multiplied by the reciprocal of the variance of the corresponding observation, the resulting weighted least squares normal equations are

but the response is affected by a second variable x2 such that the true regression function is E1Y 2

Table 3.11: PKZIP Encryption encrypt plaintext byte p result is ciphertext byte c given current X ,Y , Z k = getKeystreamByte(2) c=p@k update(X, y,2, ) P The functions getKeystreamByte and update are defined in Tables 3.12 and 3.13, respectively. T h e decryption process is easily derived from these encryption routines. Table 3.12: PKZIP getKeystreamByte getKeystrciamByte( 2)

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

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

2 2 i

// // //

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.

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

Understanding and Controlling Exposure Using aperture Using shutter speed Using ISO Understanding and Setting the White Balance Choosing a preset white balance Setting a custom white balance Setting a white balance shift About My Colors Color Accent Color Swap Custom Colors Using the Onboard Flash Flash Exposure Compensation Shutter sync Red-Eye Reduction lamp Safety FE ..

y CE y = 00010101 @ 00000101 = 00010000.

11-4

SOME COMMENTS ON USES OF REGRESSION (CD ONLY)

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