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where we have assumed that the expected value of the error term is zero; that is E( ) 0. The parameter 0 is the intercept of the plane. We sometimes call 1 and 2 partial regression coefficients, because 1 measures the expected change in Y per unit change in x1 when x2 is held constant, and 2 measures the expected change in Y per unit change in x2 when x1 is held constant. Figure 12-1(b) shows a contour plot of the regression model that is, lines of constant E(Y ) as a function of x1 and x2. Notice that the contour lines in this plot are straight lines.

Sometimes a partition of the question into successive picks is an easier method to solve the problem. EXAMPLE 2-18 A day s production of 850 manufactured parts contains 50 parts that do not meet customer requirements. Two parts are selected randomly without replacement from the batch. What is the probability that the second part is defective given that the rst part is defective Let A denote the event that the rst part selected is defective, and let B denote the event that the second part selected is defective. The probability needed can be expressed as P1B A2. If the rst part is defective, prior to selecting the second part, the batch contains 849 parts, of which 49 are defective, therefore P1B A2 EXAMPLE 2-19 49 849

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Formulas for converting WAIS-III sums of scaled scores of subtests composing various clusters to standard scores having a mean of 100 and SD of 15 Formula Component Subtests

M. Sachs et al: Executable Trading-Partner Agreements in Electronic Commerce, IBM T. J. Watson Research Center, January, 2000 Intrusion Detection Working Group: Intrusion Detection Message Exchange Requirements, Draft October 22, 2002

Continuing the previous example, if three parts are selected at random, what is the probability that the rst two are defective and the third is not defective This event can be described in shorthand notation as simply P(ddn). We have P1ddn2 50 850 49 849 800 848 0.0032

Bannatyne Categories Verbal Conceptualization Spatial Sequential Acquired Knowledge Horn Groupings Fluid Intelligence Crystallized Intelligence Broad Visualization Broad Speediness Short-Term Memory 1.1 Xss + 34 1.2 Xss + 40 1.5 Xss + 40 1.9 Xss + 43 Same as WMI MR + BD + OA + S + PA + A I + V + C + S + PA MR + BD + OA + PC CD + SS + OA A + DS + LNS 1.8 Xss + 46 1.5 Xss + 40 1.6 Xss + 36 1.9 Xss + 43 V+S+C MR + BD + OA + PC A + DS + CD + LNS I+A+V

The third term is obtained as follows. After the rst two parts are selected, there are 848 remaining. Of the remaining parts, 800 are not defective. In this example, it is easy to obtain the solution with a conditional probability for each selection. EXERCISES FOR SECTION 2-4

2-57. Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized as follows: shock resistance high low 70 9 16 5 Let A denote the event that a sample has excellent surface nish, and let B denote the event that a sample has excellent length. Determine: (a) P1A2 (b) P1B2 (c) P1A B2 (d) P1B A2 (e) If the selected part has excellent surface nish, what is the probability that the length is excellent (f) If the selected part has good length, what is the probability that the surface nish is excellent 2-59. The analysis of shafts for a compressor is summarized by conformance to speci cations: roundness conforms yes no 345 5 12 8

[IDWG02]

NOTE: Each formula applies to the entire 16- to 89-year age range. Xss equals the sum of the scaled scores for the subtests in each grouping.

Let A denote the event that a disk has high shock resistance, and let B denote the event that a disk has high scratch resistance. Determine the following probabilities: (a) P(A) (b) P(B) (c) P1A B2 (d) P1B A2 2-58. Samples of a cast aluminum part are classified on the basis of surface nish (in microinches) and length measurements. The results of 100 parts are summarized as follows: length excellent 80 10

the Crystallized subtest grouping score in Horn s system is 1.2 Xss + 40. Xss equals the person s sum of scaled scores on the subtests that make up each category. For the Crystallized grouping, these subtests are Information, Vocabulary, Comprehension, Similarities, and Picture Arrangement. For example, Brandon Y. earned the following scaled scores on these five tasks: 11, 16, 12, 11, and 15, respectively. The sum of these scores equals 65; hence, substitute 65 for Xss in the formula. 1.2 (65) + 40 = 78 + 40 = 118. This value becomes Brandon s Crystallized standard score.

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