These are my questions.
a) [6 pts] Let SSA denote the sum of squares for brand of detergent, SS; denote the sum of squares for temperature, and SSAB denote the sum of squares for the interaction. Under the two-way ANOVA model, and if brand doesn't have an effect, then what distribution does B/ 2 follows? If the distribution involves degrees of freedom, provide the explicit number of degrees of freedom. b) [10 pts] Complete the following two-way ANOVA model (filled cells do not require entries): Source Degrees of Sum of squares Mean sum of F freedom squares Brand 60 Temperature 120 Interaction 20 Error 162 c) [9 pts] Does the ability of a brand of detergent to remove dirt depend on the temperature at which clothes are washed? Use a significance level a = 0.05. Provide the null and alternative hypothesis (with the null hypothesis expressed in mathematical form), report the test statistics, the p-value, your decision and state your conclusion. In all calculations, round numbers to two decimal digits.Suppose that you purchased two brands of laundry detergents ("Cleaner" and "Tide") and choose three different temperature levels ("cold", "warm" and "hot"). For each of the 6 combinations, you measured the amount of dirt removed 10 times.TABLE 3 Percentiles of the x2 Distribution-Values of Xp Corresponding to P P Xp df X:005 X.01 X.025 X.os X.10 X.30 X.95 X.975 X.90 X.095 000039 .00016 00098 0039 0158 2.71 3.84 5.02 6.63 7.88 .0100 0201 0506 1026 .2107 4.61 5.99 7.38 9.21 10.60 AWNE .0717 115 216 352 .584 6.25 7.81 9.3 11.34 12.84 .207 297 .484 .711 1.064 7.78 9.49 11.14 13.28 14.86 .412 .554 831 1.15 1.61 9.24 11.07 12.83 15.09 16.75 676 .872 1.24 1.64 2.20 10.64 12.59 14.45 16.81 18.55 .989 1.24 1.69 2.17 2.83 12.02 14.07 16.01 18.48 20.28 10 00 NO 1.34 1.65 2.18 2.73 3.49 13.36 15.51 17.53 20.09 21.96 1.73 2.09 2.70 3.33 4.17 14.68 16.92 19.02 21.67 23.59 10 2.16 2.56 3.25 3.94 4.87 15.99 18.31 20.48 23.21 25.19 11 2.60 3.05 3.82 4.57 5.58 17.28 19.68 21.92 24.73 26.76 12 3.07 3.57 4.40 5.23 6.30 18.55 21.03 23.34 26.22 28.30 13 3.57 4.11 5.01 5.89 7.04 19.81 22.36 24.74 27.69 29.82 14 4.07 4.66 5.63 6.57 7.79 21.06 23.68 26.12 29.14 31.32 15 4.60 5.23 6.26 7.26 8.55 22.31 25.00 27.49 30.58 32.80 16 5.14 5.81 6.91 7.96 9.31 23.54 26.30 28.85 32.00 34.27 18 6.26 7.01 8.23 9.39 10.86 25.99 28.87 31.53 34.81 37.16 20 7.43 8.26 9.59 10.85 12.44 28.41 31.41 34.17 37.57 40.00 24 9.89 10.86 12.40 13.85 15.66 33.20 36.42 39.36 42.98 45.56 30 13.79 14.95 16.79 18.49 20.60 40.26 43.77 46.98 50.89 53.67 40 20.71 22.16 24.43 26.51 29.05 51.81 55.76 59.34 63.69 66.77 60 35.53 37.48 40.48 43.19 46.46 74.40 79.08 83.30 88.38 91.95 120 83.85 86.92 91.58 95.70 100.62 140.23 |146.57 152.21 158.95 | 163.64Percentiles of the t distribution Values of tr corresponding to P A .. unnnnmnr 1 [.316 6.314 [2.106 31.821 63.651 2 .239 .611 [.1151 1.336 2.920 4.303 6965 9.925 3 .211 .534 .913 1.633 2.353 3.132 4.541 5.341 4 .211 .569 .941 1.533 2.132 2.116 3.141 4.604 5 .261 .559 .920 1.416- 21115 2.511 3.365 4.1132 6 .265 .553 .9116 1.440 1.943 2.441 3.143 3.101 1 .263 .549 .396 1.4\" 1.395 2.365 2993 3.499 3 .262 .546 .339 1.391 1.360 2.306 2.396 3.355 9 .261 .543 .883 1.353 1.233 2.262 2.321 3.250 It] .260 .542 .319 1.312 1.812 2.223 2.164 3.169 11 .260 .540 .316 1.363 1.196 2.201 2.113 3.106 12 .259 .539 .313 1.356 1.132 2.119 2.631 3.055 13 .259 .533 .310 1.350 1.111 2.160 2.650 3.012 14 .253 .531 .3611 1.345 1.161 2.14.5 2.624 2.911 15 .253 .536 .366 1.341 1.153 2.131 2.602 2.941 16 .253 .535 .365 1.331 1.146 2.120 2.533 2.921 11 .251 .534 .363 1.333 1.140 2.110 2.561 2.393 13 .251 .534 .862 1 3311 1.134 2.101 2.552 2.813 19 .251 .533 .861 1 32.8 1.129 2.013 2.539 2.861 20 .251 .533 .3611 1.325 1.125 2.0116 2.523 2.345 21 .251 .532 .359 1.323 1.121 2.030 2.513 2.331 22 .36 .532 .353 1.321 1.111 2.014 2.503 2. 319 23 .256 .532 .358 1.3l9 1.1l4 2.069 2.500 2. 301 24 .256 .53l .351 1.313 1.111 2.064 2.492 2.191 25 .256 .53l .356 1.3-[6 1.106 2.060 2485 2.131 26 .256 .53l .356 1.3-[5 1.106 2. I166 1.419 2.119 21 .256 .53l .855 1.314 1.103 2. I152 2.413 2.111 23 .256 .530 .355 1.313 1.101 2.043 2.461 2.163 29 .256 .531 .354 1.311 1.699 2 .045 2.462 2.156 30 .256 .530 .354 1.3"] 1.691 2.042 2.451 2.150 40 .255 .529 .35] 1.30.] 1.664 2.D2l 2.423 2.1114 60 .254 .521 .343 1.2% 1.611 2.1110 2.390 2.660 120 .254 .526 .345 1.239 1.653 1.930 2.353 2.611 on .253 .524 .342 1.232 1.645 1.960 2.326 2.516 TABLE 5 Percentiles of the F Distribution: F.95 (m1, n2) (Continued) 71 = degrees of freedom for numerator 10 12 15 20 120 DO 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 241.9 243.9 245.9 248.0 249.1 250.1 251.1 252.2 253.3 254.3 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.41 19.43 19.45 19.45 19.46 19.47 19.48 19.49 19.50 AWNE 10.13 9.55 9.28 9.12 9.01 8.9 8.89 8.85 8.81 8.79 8.74 8.70 8.66 8.64 8.6 8.59 8.57 8.55 8.53 7.7 6.94 5.59 6.39 6.26 6.16 6.09 6.04 6.010 5.96 5.91 5.86 5.80 5.77 5.75 5.72 5.69 5.60 5.63 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.7 4.74 4.68 4.62 4.50 4.53 4.50 4.46 4.43 4.40 4.36 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.00 3.94 3.87 3.84 3.81 3.77 3.74 3.70 3.67 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.57 3.51 3.44 3.41 3.38 3.34 3.30 3.27 3.23 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.28 3.22 3.15 3.12 3.08 3.04 3.01 2.97 2.93 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.07 3.01 2.94 2.90 2.86 2.83 2.79 2.75 2.71 1.96 1.10 3.71 3.48 3.83 3.22 3.14 3.07 3.02 2.98 2.91 2.85 2.77 2.74 2.70 2.66 2.62 2.58 2.54 4.84 3.9 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85 2.79 2.72 2.65 2.61 2.57 2.53 2.49 2.45 2.40 1.75 3.8 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 2.69 2.62 2.54 2.51 2.47 2.43 2.38 2.34 2.30 /12 = degrees of freedom for denominator 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67 2.60 2.53 2.46 2.42 2.38 2.34 2.30 2.25 2.21 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60 2.53 2.46 2.39 2.35 2.31 2.27 2.22 2.18 2.13 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 2.48 2.40 2.33 2.29 2.25 2.20 2.16 2.11 2.07 1.49 3.6. 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49 2.42 2.35 2.28 2.24 2.19 2.15 2.11 2.06 2.01 4.45 3.5 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2.45 2.38 2.31 2.23 2.19 2.15 2.10 2.06 2.01 1.96 1.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41 2.34 2.27 2.19 2.15 2.11 2.06 2.02 1.97 1.92 .38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2.38 2.31 2.23 2.16 2.11 2.07 2.03 1.98 1.93 1.88 1.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35 2.28 2.20 2.12 2.08 2.04 1.99 1.95 1.90 1.84 1.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 2.25 2.18 2.10 2.05 2.01 1.96 1.92 1.87 1.81 .30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34 2.30 2.23 2.15 2.07 2.03 1.98 1.94 1.89 1.84 1.78 1.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32 2.27 2.20 2.13 2.05 2.01 1.96 1.91 1.86 1.81 1.76 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25 2.18 2.11 2.03 1.98 1.94 1.89 1.84 1.79 1.73 4.24 3.3 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24 2.16 2.09 2.01 1.90 1.92 1.87 1.82 1.77 1.71 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27 2.22 2.15 2.07 1.99 1.95 1.90 1.85 1.80 1.75 1.69 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25 2.20 2.13 2.06 1.97 1.93 1.88 1.84 1.79 1.73 1.67 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24 2.19 2.12 2.04 1.96 1.91 1.87 1.82 1.77 1.71 1.65 4.18 3.3 2.93 2.70 2.55 2.43 2.35 2.28 2.22 2.18 2.10 2.03 1.94 1.90 1.85 1.81 1.75 1.70 1.64 1.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 2.09 2.01 1.93 1.89 1.84 1.79 1.74 1.68 1.62 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08 2.00 1.92 1.84 1.79 1.74 1.69 1.64 1.58 1.51 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99 1.92 1.84 1.75 1.70 1.65 1.59 1.53 1.47 1.39 3.92 3.07 2.68 2.45 2.29 2.17 2.09 2.02 1.96 1.91 1.83 1.75 1.66 1.61 1.55 1.50 1.43 1.35 1.25 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88 1.83 1.75 1.67 1.57 1.52 1.46 1.39 1.32 1.22 1.00
