1、*KinM*DS AIEE C _ .IUL June 1958 also with SS = SS/n. The quantity S, which can now be written simply as VSS Error/(N-2) is a measure of the average spread in the Ys for a given value of X.* The variance of the predicted mean value Y for a given value X is given by: ri (X-X)2 - Sx = S8 - + - (9) LN
2、2(X-Xr -I If curves are drawn about the regression line (1) at a distance of Y tcSy, as shown in Figure 1F-4, the resulting band can be interpreted as confidence limits on the predicted mean value Y. The exact value of the factor tc depends on the number of tests minus two, or N-2, and on the desire
3、d confidence level which has been tabulated in Table lF-4b. This table gives a par tial tabulation of the tc values (see Hald3, p. 388). For 95 percent confidence, and for N 28, tc is (to one decimal place) equal to 2.0. Work Sheet lF-4c, applying to both grouped and paired data, indicates the compu
4、tations involved. lF-4c Linearity of Regression The validity of the regression analysis given above rests on the assumption that the relationship between Y and X is linear. It is possible to make a statistical test for the correctness of this assumption. In case the test for linearity is negative, a
5、 curvilinear regres sion might be used. Alternatively, the knowledge of such non-linearity might be used to make an intuitive allowance in the prediction limits given in the preced ing paragraphs. The test for linearity is especially simple for case (b), when the same number of samples are tested at
6、 each of k temperatures. Only this case will be dis cussed. The following SS are introduced: SS Between Xs and Y,s = n2(Z-“Z)* SS Non-Linear = SS Between Xs and Ys SS Regression SS Within Xs and Ys = SS Total-SS Be tween To test for linearity, the quantity: k(N-l) SS Non-Linear (10) F = k-2 SS Withi
7、n or, k(n-l) 2Z2-Z2Z-b2WZ (10a) F = k“2 * 2Y2-2Z2 n is computed and compared with the tabulated critical significance values of Fc of the F distribution (Table lF-4c), for k-2 and k(n-l) degrees of freedom. If the computed value is less than the critical value, then the regression is presumed to be
8、linear. Work Sheet lF-4d, Items 47 to 54 illustrate this test. *The correlation coefficient r can, if desired, be computed from VSS Regression/SS Total. lF-4d Comparing Two Regressions The results on a particular component type tested may be compared with those of a previous type. The most common si
9、tuation would be one in which it is desired to compare mean lives under the same rated temperature X as predicted from sample tests. Using the subscripts A and B to represent respectively the two sample types, the difference in predicted mean life, YA YB, is divided by the combined standard error an
10、d the result denoted by v as shown in Equation (11): YA-YB The final results of all the tests may be summarized by a graph showing the regression line in terms of life and temperature and the 95 percent confidence limits of this line. References (1) Dakin, T. W., Electrical Insulation Deterioration
11、Treated as a Chemical Rate Phenomenon, AIEE Transaction, Volume 67, Part 1, 1948, pages 113-122. (2) Horton, W. H., A Statistical Method for Predicting Insulation Life From Experimental Data, POWER APPARATUS AND SYSTEMS, AIEE, Volume 75, No. 25, August, 1956, pages 520-527. (3) Hald, A., Statistical
12、 Theory With Engineering Ap plications, 1952, pages 129-136, John Wiley and Sons, New York. 7 Table lF-4a DATA FOR EXAMPLE CALCULATIONS* Measurements of Insulation Life for Motorettes Operated at Designated Temperatures Test Temperature C X = l/K 125 0.002513 140 0.002421 160 0.002309 180 0.002208 G
13、roup A Number of Motorettes 6 1 2 1 9 1 1 6 3 1 7 2 Hours 4116 5292 6468 7644 1680 2352 288 672 864 216 264 312 Y= 1 Log Hours 3.61448 3.72362 3.81077 3.88332 3.22531 3.37144 2.45939 2.82737 2.93651 2.33445 2.42160 2.49415 Group B Number of Motorettes 2 7 1 6 4 7 3 2 3 3 1 1 Hours 2940 4116 5292 168
14、0 2352 672 864 216 264 312 360 408 Y = Log Hours ! j 3.46835 3.61448 3.72362 3.22531 3.37144 2.82737 2.93651 2.33445 2.42160 2.49415 2.55630 2.61066 *Data taken from “Test Procedure for Evaluation of Systems of Insulating Materials for Random-Wound Electric Machinery“ AIEE No. 510Revision of Origina
15、l AIEE No. lc, 4/11/56. 8 Table lF-4b t-TABLE 0.05 Probability N-2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30 40 60 120 oo Level (two-sided) tc 12.706 4.303 3.182 2.776 2.571 2.477 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.0
16、74 2.069 2.064 2.060 2.042 2.021 2.000 1.980 1.960 9 Table lF-4c F-TABLE 0.05 Probability Level . k-2 k (n-l) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30 40 60 120 00 1 161.4 18.51 10.13 7.71 6.61 5.99 5.59 5.32 5.12 4.96 4.84 4.75 4.67 4.60 4.54 4.49 4.45 4.41 4.38 4.35 4.3
17、2 4.30 4.28 4.26 4.24 4.17 4.08 4.00 3.92 3.84 2 199.5 19.0 9.55 6.94 5.79 5.14 4.74 4.46 4.26 4.10 3.98 3.88 3.80 3.74 3.68 3.63 3.59 3.55 3.52 3.49 3.47 3.44 3.42 3.40 3.38 3.32 3.23 3.15 3.07 3.00 3 215.7 19.16 9.28 6.59 5.41 4.76 4.35 4.07 3.86 3.71 3.59 3.49 3.41 3.34 3.29 3.24 3.20 3.15 3.13 3
18、.10 3.07 3.05 3.03 3.01 2.99 2.92 2.84 2.76 2.68 2.60 4 224.6 19.25 9.12 6.39 5.19 4.53 4.12 3.84 3.63 3.48 3.36 3.26 3.18 3.11 3.06 3.01 2.96 2.93 2.90 2.87 2.84 2.82 2.80 2.78 2.76 2.69 2.61 2.53 2.45 2.37 , 5 230.2 19.30 9.01 6.26 5.05 4.39 3.97 3.69 3.48 3.33 3.20 3.11 3.02 2.96 2.90 2.85 2.81 2
19、.77 2.74 2.71 2.68 2.66 2.64 2.62 2.60 2.53 2.45 2.37 2.29 2.21 6 234.0 19.33 8.94 6.16 4.95 4.28 3.87 3.58 3.37 3.22 3.09 3.00 2.92 2.85 2.79 2.74 2.70 2.66 2.63 2.60 2.57 2.55 2.53 2.51 2.49 2.42 2.34 2.25 2.18 2.10 8 238.9 19.37 8.84 6.04 4.82 4.15 3.73 3.44 3.23 3.07 2.95 2.85 2.77 2.70 2.64 2.5
20、9 2.55 2.51 2.48 2.45 2.42 2.40 2.38 2.36 2.34 2.27 2.18 2.10 2.02 1.94 12 243.9 19.41 8.74 5.91 4.68 4.00 3.57 3.28 3.07 2.91 2.79 2.69 2.60 2.53 2.48 2.42 2.38 2.34 2.31 2.28 2.25 2.23 2.20 2.18 2.16 2.09 2.00 1.92 1.83 1.75 24 249.0 19.45 8.64 5.77 4.53 3.84 3.41 3.12 2.90 2.74 2.61 2.50 2.42 2.35 2.29 2.24 2.19 2.15 2.11 2.08 2.05 2.03 2.00 1.98 1.96 1.89 1.79 1.70 1.61 1.52 10
