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(d) Find a 95% prediction interval on steam usage when temperature is 55 F . Explain why this interval is wider than the interval in part (c). 11-36. Exercise 11-7 presented gasoline mileage performance for 20 cars, along with information about the engine displacement. Find a 95% con dence interval on each of the following: (a) Slope (b) Intercept (c) Mean highway gasoline mileage when the engine displacement is x 150 in3 (d) Construct a 95% prediction interval on highway gasoline mileage when the engine displacement is x 150 in3. 11-37. Consider the data in Exercise 11-8 on y green liquor Na2S concentration and x production in a paper mill. Find a 99% con dence interval on each of the following: (a) 1 (b) 0 (c) Mean Na2S concentration when production x 910 tons day (d) Find a 99% prediction interval on Na2S concentration when x 910 tons day. 11-38. Exercise 11-9 presented data on y blood pressure rise and x sound pressure level. Find a 95% con dence interval on each of the following: (a) 1 (b) 0

(c) Mean blood pressure rise when the sound pressure level is 85 decibals (d) Find a 95% prediction interval on blood pressure rise when the sound pressure level is 85 decibals. 11-39. Refer to the data in Exercise 11-10 on y wear volume of mild steel and x oil viscosity. Find a 95% con dence interval on each of the following: (a) Intercept (b) Slope (c) Mean wear when oil viscosity x 30 11-40. Exercise 11-11 presented data on chloride concentration y and roadway area x on watersheds in central Rhode Island. Find a 99% con dence interval on each of the following: (a) 1 (b) 0 (c) Mean chloride concentration when roadway area x 1.0% (d) Find a 99% prediction interval on chloride concentration when roadway area x 1.0%. 11-41. Refer to the data in Exercise 11-12 on rocket motor shear strength y and propellant age x. Find a 95% con dence interval on each of the following: (a) Slope 1 (b) Intercept 0 (c) Mean shear strength when age x 20 weeks (d) Find a 95% prediction interval on shear strength when age x 20 weeks.

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As mentioned above, the fundamental problem with ORYX is that it attcmpts to generate a byte of keystream at each iteration. While this makes for an efficient cipher, it exposes far too much of the internal state to thc attackcr. Can we modify ORYX so that it is more secure If, at each itcration, we output a single bit instcad of a byte, that would probably improve the security

11-8

Fitting a regression model requires several assumptions. Estimation of the model parameters requires the assumption that the errors are uncorrelated random variables with mean zero and constant variance. Tests of hypotheses and interval estimation require that the errors be normally distributed. In addition, we assume that the order of the model is correct; that is, if we t a simple linear regression model, we are assuming that the phenomenon actually behaves in a linear or rst-order manner. The analyst should always consider the validity of these assumptions to be doubtful and conduct analyses to examine the adequacy of the model that has been tentatively entertained. In this section we discuss methods useful in this respect.

significantly. For example, suppose we were to compute each keystream bit as keystreamBit = s ( X )@ s ( L [ H ( A ) ] ) s ( L [ H ( B ) ] ) , @ where s selects, say, the high (rightmost) bit of a word or byte. Provided that a complete iteration occurs between each keystreamBit computation, this modification would frustrate the attack discussed above-even if much more known plaintext were available-since the number of candidates to be considered would grow rapidly, instead of being reduced at each iteration. However, this modification would make the cipher eight times slower, which is almost certainly not practical for its intended application. Also, other attacks on this secure version of ORYX would need to be considered.

11-8.1

The residuals from a regression model are ei yi yi, i 1, 2, p , n , where yi is an actual observation and yi is the corresponding tted value from the regression model. Analysis of the residuals is frequently helpful in checking the assumption that the errors are approximately normally distributed with constant variance, and in determining whether additional terms in the model would be useful. As an approximate check of normality, the experimenter can construct a frequency histogram of the residuals or a normal probability plot of residuals. Many computer programs will produce a normal probability plot of residuals, and since the sample sizes in regression are often too small for a histogram to be meaningful, the normal probability plotting method

Figure 11-9 Patterns for residual plots. (a) satisfactory, (b) funnel, (c) double bow, (d) nonlinear. [Adapted from Montgomery, Peck, and Vining (2001).]

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