J d 7. Do the Average score of a mathematics aptitude test differ by school region?...

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J d 7. Do the Average score of a mathematics aptitude test differ by school region? Data were collected from 3 schools from Newton Middle School, Lancaster Middle School, and Fitchburg Middle School. The results of the analysis are given below: < School Newton Fitchburg Lancaster J J Descriptives < Between Groups Within Groups Total a. Sum of Squares 77,760* 8,280* 86,040* Ne 35* 28 40* ANOVA < Df 2 92* 94 < Mean 89* 67 82 Mean Square* 38,880* 90* e F 432* e e Standard deviation 6.3 7.1 5.2 Sig. 0.0025* e Assume the data is normally distributed. ANOVA requires also that the standard deviations be equal. Using the > 2 or < 2 rule, please state if ANOVA meets this assumption. Explain your answer. < b. In order to test if the mathematical aptitude tests differ by School, state the null and alternative hypothesis. < C. At a level of significance of a= 0.05, What is your conclusion? Explain your findings. P Table A: Standardized z-score chart for positive z-scores < STANDARD STATISTICAL TABLES 1. Areas under the Normal Distribution The table gives the cumulative probability up to the standardised normal value z i.e. 2 P[ Z < 2 ] Z CEEEE EEEE! 88::: ::::: 0.0 0.5000 0.1 0.5398 0.2 0.5793 0.3 0.6179 0.6554 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 0.00 1.9 -0 1 exp(-2) dz 27 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.5040 0.5080 0.5120 0.5159 0.5438 0.5478 0.5517 0.5557 0.5832 0.5871 0.5910 0.5948 0.6217 0.6255 0.6293 0.6331 0.6591 0.6628 0.6664 0.8413 0.8438 0.8461 0.8643 0.8665 0.8686 0.8849 0.8869 0.8888 0.9032 0.9049 0.9066 0.9192 0.9207 0.9222 0 0.8485 0.8508 0.8708 0.8729 0.8907 0.8925 0.9082 0.9099 0.9236 0.9251 P[ Z < Z ] 0.9382 9332 0.9345 0.9357 0.931 0.9452 0.9463 0.9474 0.9484 0.9495 0.9554 0.9564 0.9573 0.9582 0.9591 0.9641 0.9649 0.9656 0.9664 0.9671 0.9713 0.9719 0.9726 0.9732 0.9738 Z 0.08 0.5199 0.5596 0.5987 0.6026 0.6064 0.6103 0.6368 0.6406 0.6443 0.6480 0.6700 0.6736 0.6772 0.6808 0.6844 0.6915 0.6950 0.6985 0.7019 0.7054 0.7257 0.7291 0.7324 0.7357 0.7389 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8078 0.8106 0.8340 0.8365 0.5239 0.5279 0.5319 0.5359 0.5636 0.5675 0.5714 0.5753 0.7088 0.7123 0.7157 0.7190 0.7224 0.7422 0.7454 0.7486 0.7517 0.7549 0.7854 0.8531 0.8554 0.8577 0.8599 0.8749 0.8770 0.8790 0.8804 0.8944 0.8962 0.8980 0.8997 0.9147 0.9162 0.9115 0.9131 0.9265 0.9279 0.9292 0.9306 9394 9406 0.9505 0.9515 0.9599 0.9608 0.9678 0.9686 0.9693 0.9699 0.9744 0.9750 0.9756 0.9761 9418 0.9 0.09 0.9525 0.9535 0.9616 0.9625 0.6141 0.6517 0.6879 0.8133 0.8389 0.8621 0.8830 0.9015 0.9177 0.9319 0.9441 0.9545 0.9633 0.9706 0.9767 2.0 0.9773 0.9778 0.9783 0.9788 0.9793 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 2.2 0.9861 0.9865 0.9868 0.9871 2.3 0.9893 0.9896 0.9898 0.9901 2.4 0.9918 0.9920 0.9922 0.9924 0.9798 0.9803 0.9808 0.9812 0.9817 0.9842 0.9846 0.9850 0.9854 0.9857 0.9874 0.9878 0.9881 0.9884 0.9887 0.9904 0.9906 0.9909 0.9911 0.9913 0.9931 0.9932 0.9934 0.9927 0.9929 NNNNN 0.9945 0.9946 2.5 0.9938 0.9940 0.9941 0.9943 0.9948 0.9949 0.9951 0.9952 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9980 0.9980 0.9981 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 2 P 0.9890 0.9916 0.9936 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 0.9986 0.9990 0.9993 0.9995 0.9997 0.9998 0.9998 0.9999 0.9999 1.0000 df 1 2 3 5 6 7 8 9 10 11 12 13 12245 X305 0.000 0.010 0.072 0.207 0412 0.676 0.989 1344 1.735 2.156 2.603 3.074 3.565 4.075 4.601 16 5.142 17 5.697 18 6.265 19 6844 20 7.434 X300 0.000 0.020 0.115 0.297 0.554 0.872 1.239 1.646 2.088 2.558 3.053 3.571 4.107 4.660 5.229 5.812 6.408 7.015 7.633 8.260 Chi-Square Distribution Table The shaded area is equal to a for x = x. X-975 0.001 0.051 0.216 0.484 0.831 1.237 1.690 2.180 2.700 3.247 3.816 4.404 5.009 5.629 6.262 6.908 7.564 0 8.231 8.907 9.591 X-950 0.004 0.103 0.352 0.711 1.145 1.635 2.167 2.733 3.325 3.940 4.575 5.226 5.892 X3900 0.016 0.211 0.584 1.064 1.610 2.204 2.833 3.490 4.168 4.865 5.578 6.304 7.042 X100 X050 2.706 3.841 4.605 5.991 6.251 7.815 9.348 11.345 7.779 9.488 11.143 13.277 9.236 11.070 12.833 15.086 10.645 14.449 12.592 14.067 16.013 12.017 13.362 15.507 14.684 16.919 15.987 18.307 19.675 18.549 21.026 17.275 19.812 22.362 7.790 8.547 6.571 7.261 7.962 8.672 9.390 10.865 25.989 11.651 27.204 10.117 10851 12.443 28.412 X025 5.024 7.378 21.064 26.119 23.685 22.307 24.996 27.488 9.312 23.542 10.085 24.769 X010 6.635 9.210 28.845 26.296 27.587 30.191 28.869 31.526 30.144 32.852 31.410 34.170 16812 18.475 17.535 20.090 23.589 19.023 21.666 20.483 23.209 21.920 24.725 25.188 26.757 23.337 26.217 28.300 24.736 27.688 29.819 31.319 32.801 29.141 30.578 005 32.000 33.409 7.879 10.597 12.838 14.860 16.750 18.548 20.278 21.955 34.267 35.718 34.805 37.156 36.191 38.582 37.566 39.997 35.172 38.076 8.897 10.283 11.591 9.542 10.982 12.338 10.196 11.689 13.091 10.856 12.401 13848 13.120 14.611 15.379 13.240 29.615 32.671 14.041 30.813 14.848 32.007 15.659 33.196 36.415 16.473 34.382 37.652 38.885 40.113 43.195 10.520 11.524 17.292 27 12.198 13.844 11.808 12.879 14.573 16.151 15.308 16.928 18.939 18.114 28 12.461 13.565 41.337 44.461 29 17.708 19.768 30 18.493 20.599 24.433 26.509 29.051 40 20.707 22.164 50 27.991 29.707 32.357 34.764 35.534 37.485 40.482 43.188 43.275 45.442 48.758 51.739 55.329 51.172 53.540 57.153 60.391 64.278 37.689 46.459 74.397 60 85.527 96.578 73.291 107.565 113.145 118.136 124.116 82.358 118.498 124.342 129.561 135.807 21 8.034 22 8.643 23 9.260 24 9.886 25 26 11.160 70 80 13.121 13.787 14.256 16.047 14.953 16.791 90 59.196 61.754 65.647 100 67.328 70.065 74.222 69.126 77.929 36.741 37.916 39.087 40.256 35.479 38.932 41.401 33.924 36.781 40.289 42.796 41.638 44.181 39.364 42.980 45.559 40.646 44314 46.928 41.923 45.642 48.290 46.963 48.278 42.557 45.722 49.588 43.773 46.979 50.892 59.342 63.691 76.154 83.298 88.379 95.023 100.425 106.629 112.329 51.805 55.758 63.167 67.505 71.420 79.082 90.531 101 879 49.645 50.993 52.336 53.672 66.766 79.490 91.952 104.215 116.321 128.299 140.169 Table A Continued: Standardized Z-score Chart for negative z-scores. < Area- Table IV Standard Normal Distribution .03 04 .05 z .00 .01 .02 .06 .07 .08 -3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002 -3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003 -3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005 -3.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007 -3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010 -2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014 -2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019 -2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026 -2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036 -2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048 -2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 -2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 -2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110 -2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143 -2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183 0.0262 0.0256 0.0250 0.0329 0.0064 0.0084 -1.9 0.0287 0.0244 0.0239 0.0281 0.0274 0.0268 0.0336 -1.8 0.0359 0.0351 0.0344 0.0322 0.0314 0.0307 0.0301 0.1190 0.1170 -1.0 0.1401 0.1379 0.1635 0.1611 -1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 -1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 -1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 -1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681 -1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 -1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985 -1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 -0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 -0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867 -0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148 -0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451 -0.5 0.3085 03050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776 -0.4 0.3446 03409 03372 0.3336 0.3300 0.3264 0.3228 03192 0.3156 0.3121 -0.3 0.3821 03783 03745 0.3707 0.3669 0.3632 0.3594 0.3557 03520 0.3483 -0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 03897 0.3859 -0.1 0.4602 0.4562 0,4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247 -0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4761 0.4721 0.4681 0.4641 0.4801 09 0.0233 0.0294 0.0367 0.0455 0.0559 Table C: T-distribution Chart t Table cum. prob one-tail two-tails df 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 t.75 t so 0.50 0.25 1.00 0.50 1.000 1.376 1.963 3.078 6.314 12.71 31.82 63.66 318.31 6.965 9.925 22.327 0.816 1.061 0.765 0.978 0.741 7.173 0.000 0.727 1.386 1.886 2.920 4.303 1.250 1.638 2.353 3.182 4.541 5.841 10.215 0.941 1.190 1.533 2.132 2.776 3.747 4.604 1.476 2.015 2.571 3.365 1.440 1.943 2.447 3.143 1.895 2.365 2.998 2.306 4.032 5.893 0.920 1.156 0.906 1.134 0.896 1.119 0.000 0.718 3.707 5.208 0.000 0.711 1.415 3.499 4.785 0.000 0.706 0.889 1.397 1.860 2.896 3.355 4.501 1.108 0.883 1.100 1.383 1.833 2.262 2.821 3.250 4.297 4.781 1.812 2.228 4.144 4.587 0.000 0.703 0.000 0.700 0.879 1.093 1.372 0.000 0.697 0.876 1.088 1.363 0.873 1.083 1.356 1.782 1.796 2.201 3.106 4.025 4.437 0.000 0.695 2.179 3.055 3.930 4.318 0.000 0.694 1.079 1.350 1.771 3.012 3.852 4.221 0.870 0.868 1.076 0.000 0.692 1.345 1.761 2.624 2.977 3.787 4.140 0.000 0.691 1.753 3.733 4.073 0.000 0.690 2.921 3.686 4.015 0.000 3.819 2.074 2.508 2.819 3.505 3.792 2.069 2.500 2.807 3.485 3.768 2.064 2.797 3.467 3.745 2.492 2.485 2.787 3.725 0.000 0.684 0.866 1.074 1.341 2.602 2.947 0.865 1.071 1.337 1.746 2.583 0.689 0.863 1.069 1.333 1.740 2.110 2.567 2.898 3.646 3.965 0.000 0.688 0.862 1.067 1.330 1.734 2.101 2.552 2.878 3.610 3.922 0.000 0.688 0.861 1.066 1.328 1.729 2.093 2.539 2.861 3.579 3.883 0.000 0.687 0.860 1.064 1.325 1.725 2.086 2.528 2.845 3.552 3.850 0.000 0.686 0.859 1.063 1.323 1.721 2.080 2.518 2.831 3.527 0.000 0.686 0.858 1.061 1.321 1.717 0.000 0.685 0.858 1.060 1.319 1.714 0.000 0.685 0.857 1.059 1.318 1.711 0.000 0.684 0.856 1.058 1.316 1.708 2.060 3.450 0.856 1.058 1.315 1.706 2.056 2.479 2.779 3.435 0.855 1.314 1.703 2.052 2.473 2.771 3.421 0.855 1.056 1.313 1.701 2.048 2.467 2.763 3.408 0.854 1.055 1.311 1.699 1.310 1.697 2.042 2.457 2.750 1.303 1.684 2.021 2.423 1.296 1.671 2.000 2.390 80 1.292 1.664 1.990 2.374 100 0.000 0.677 0.845 1.042 1.290 1.660 1.984 2.364 2.626 3.174 1000 0.000 1.037 1.282 1.646 1.962 2.330 2.581 3.098 1.036 1.282 1.645 1.960 2.326 2.576 3.090 3.291 80% 90% 95% 98% 99% 99.8% 99.9% Confidence Level 0.000 0.684 1.057 28 0.000 0.683 29 2.045 2.462 2.756 3.396 30 3.385 40 0.000 0.683 0.000 0.683 0.854 1.055 0.000 0.681 0.851 1.050 0.000 0.679 0.848 1.045 1.043 2.704 3.307 60 2.660 3.232 0.000 0.678 0.846 2.639 3.195 0.675 0.842 0.842 Z 60% 70% 26 27 0.000 0.000 0.000 0.000 tso tas 0.20 0.15 0.40 0.30 0.000 0.674 0% 50% t.90 0.10 0.20 t 95 0.05 0.10 t .975 0.025 0.05 t.99 0.01 0.02 t 995 0.005 0.01 t 999 0.001 0.002 2.764 3.169 2.718 2.681 2.160 2.650 2.145 2.131 2.120 t.9995 0.0005 0.001 636.62 31.599 12.924 8.610 6.869 5.959 5.408 5.041 3.707 3.690 3.674 3.659 3.646 3.551 3.460 3.416 3.390 3.300 J d 7. Do the Average score of a mathematics aptitude test differ by school region? Data were collected from 3 schools from Newton Middle School, Lancaster Middle School, and Fitchburg Middle School. The results of the analysis are given below: < School Newton Fitchburg Lancaster J J Descriptives < Between Groups Within Groups Total a. Sum of Squares 77,760* 8,280* 86,040* Ne 35* 28 40* ANOVA < Df 2 92* 94 < Mean 89* 67 82 Mean Square* 38,880* 90* e F 432* e e Standard deviation 6.3 7.1 5.2 Sig. 0.0025* e Assume the data is normally distributed. ANOVA requires also that the standard deviations be equal. Using the > 2 or < 2 rule, please state if ANOVA meets this assumption. Explain your answer. < b. In order to test if the mathematical aptitude tests differ by School, state the null and alternative hypothesis. < C. At a level of significance of a= 0.05, What is your conclusion? Explain your findings. P Table A: Standardized z-score chart for positive z-scores < STANDARD STATISTICAL TABLES 1. Areas under the Normal Distribution The table gives the cumulative probability up to the standardised normal value z i.e. 2 P[ Z < 2 ] Z CEEEE EEEE! 88::: ::::: 0.0 0.5000 0.1 0.5398 0.2 0.5793 0.3 0.6179 0.6554 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 0.00 1.9 -0 1 exp(-2) dz 27 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.5040 0.5080 0.5120 0.5159 0.5438 0.5478 0.5517 0.5557 0.5832 0.5871 0.5910 0.5948 0.6217 0.6255 0.6293 0.6331 0.6591 0.6628 0.6664 0.8413 0.8438 0.8461 0.8643 0.8665 0.8686 0.8849 0.8869 0.8888 0.9032 0.9049 0.9066 0.9192 0.9207 0.9222 0 0.8485 0.8508 0.8708 0.8729 0.8907 0.8925 0.9082 0.9099 0.9236 0.9251 P[ Z < Z ] 0.9382 9332 0.9345 0.9357 0.931 0.9452 0.9463 0.9474 0.9484 0.9495 0.9554 0.9564 0.9573 0.9582 0.9591 0.9641 0.9649 0.9656 0.9664 0.9671 0.9713 0.9719 0.9726 0.9732 0.9738 Z 0.08 0.5199 0.5596 0.5987 0.6026 0.6064 0.6103 0.6368 0.6406 0.6443 0.6480 0.6700 0.6736 0.6772 0.6808 0.6844 0.6915 0.6950 0.6985 0.7019 0.7054 0.7257 0.7291 0.7324 0.7357 0.7389 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8078 0.8106 0.8340 0.8365 0.5239 0.5279 0.5319 0.5359 0.5636 0.5675 0.5714 0.5753 0.7088 0.7123 0.7157 0.7190 0.7224 0.7422 0.7454 0.7486 0.7517 0.7549 0.7854 0.8531 0.8554 0.8577 0.8599 0.8749 0.8770 0.8790 0.8804 0.8944 0.8962 0.8980 0.8997 0.9147 0.9162 0.9115 0.9131 0.9265 0.9279 0.9292 0.9306 9394 9406 0.9505 0.9515 0.9599 0.9608 0.9678 0.9686 0.9693 0.9699 0.9744 0.9750 0.9756 0.9761 9418 0.9 0.09 0.9525 0.9535 0.9616 0.9625 0.6141 0.6517 0.6879 0.8133 0.8389 0.8621 0.8830 0.9015 0.9177 0.9319 0.9441 0.9545 0.9633 0.9706 0.9767 2.0 0.9773 0.9778 0.9783 0.9788 0.9793 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 2.2 0.9861 0.9865 0.9868 0.9871 2.3 0.9893 0.9896 0.9898 0.9901 2.4 0.9918 0.9920 0.9922 0.9924 0.9798 0.9803 0.9808 0.9812 0.9817 0.9842 0.9846 0.9850 0.9854 0.9857 0.9874 0.9878 0.9881 0.9884 0.9887 0.9904 0.9906 0.9909 0.9911 0.9913 0.9931 0.9932 0.9934 0.9927 0.9929 NNNNN 0.9945 0.9946 2.5 0.9938 0.9940 0.9941 0.9943 0.9948 0.9949 0.9951 0.9952 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9980 0.9980 0.9981 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 2 P 0.9890 0.9916 0.9936 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 0.9986 0.9990 0.9993 0.9995 0.9997 0.9998 0.9998 0.9999 0.9999 1.0000 df 1 2 3 5 6 7 8 9 10 11 12 13 12245 X305 0.000 0.010 0.072 0.207 0412 0.676 0.989 1344 1.735 2.156 2.603 3.074 3.565 4.075 4.601 16 5.142 17 5.697 18 6.265 19 6844 20 7.434 X300 0.000 0.020 0.115 0.297 0.554 0.872 1.239 1.646 2.088 2.558 3.053 3.571 4.107 4.660 5.229 5.812 6.408 7.015 7.633 8.260 Chi-Square Distribution Table The shaded area is equal to a for x = x. X-975 0.001 0.051 0.216 0.484 0.831 1.237 1.690 2.180 2.700 3.247 3.816 4.404 5.009 5.629 6.262 6.908 7.564 0 8.231 8.907 9.591 X-950 0.004 0.103 0.352 0.711 1.145 1.635 2.167 2.733 3.325 3.940 4.575 5.226 5.892 X3900 0.016 0.211 0.584 1.064 1.610 2.204 2.833 3.490 4.168 4.865 5.578 6.304 7.042 X100 X050 2.706 3.841 4.605 5.991 6.251 7.815 9.348 11.345 7.779 9.488 11.143 13.277 9.236 11.070 12.833 15.086 10.645 14.449 12.592 14.067 16.013 12.017 13.362 15.507 14.684 16.919 15.987 18.307 19.675 18.549 21.026 17.275 19.812 22.362 7.790 8.547 6.571 7.261 7.962 8.672 9.390 10.865 25.989 11.651 27.204 10.117 10851 12.443 28.412 X025 5.024 7.378 21.064 26.119 23.685 22.307 24.996 27.488 9.312 23.542 10.085 24.769 X010 6.635 9.210 28.845 26.296 27.587 30.191 28.869 31.526 30.144 32.852 31.410 34.170 16812 18.475 17.535 20.090 23.589 19.023 21.666 20.483 23.209 21.920 24.725 25.188 26.757 23.337 26.217 28.300 24.736 27.688 29.819 31.319 32.801 29.141 30.578 005 32.000 33.409 7.879 10.597 12.838 14.860 16.750 18.548 20.278 21.955 34.267 35.718 34.805 37.156 36.191 38.582 37.566 39.997 35.172 38.076 8.897 10.283 11.591 9.542 10.982 12.338 10.196 11.689 13.091 10.856 12.401 13848 13.120 14.611 15.379 13.240 29.615 32.671 14.041 30.813 14.848 32.007 15.659 33.196 36.415 16.473 34.382 37.652 38.885 40.113 43.195 10.520 11.524 17.292 27 12.198 13.844 11.808 12.879 14.573 16.151 15.308 16.928 18.939 18.114 28 12.461 13.565 41.337 44.461 29 17.708 19.768 30 18.493 20.599 24.433 26.509 29.051 40 20.707 22.164 50 27.991 29.707 32.357 34.764 35.534 37.485 40.482 43.188 43.275 45.442 48.758 51.739 55.329 51.172 53.540 57.153 60.391 64.278 37.689 46.459 74.397 60 85.527 96.578 73.291 107.565 113.145 118.136 124.116 82.358 118.498 124.342 129.561 135.807 21 8.034 22 8.643 23 9.260 24 9.886 25 26 11.160 70 80 13.121 13.787 14.256 16.047 14.953 16.791 90 59.196 61.754 65.647 100 67.328 70.065 74.222 69.126 77.929 36.741 37.916 39.087 40.256 35.479 38.932 41.401 33.924 36.781 40.289 42.796 41.638 44.181 39.364 42.980 45.559 40.646 44314 46.928 41.923 45.642 48.290 46.963 48.278 42.557 45.722 49.588 43.773 46.979 50.892 59.342 63.691 76.154 83.298 88.379 95.023 100.425 106.629 112.329 51.805 55.758 63.167 67.505 71.420 79.082 90.531 101 879 49.645 50.993 52.336 53.672 66.766 79.490 91.952 104.215 116.321 128.299 140.169 Table A Continued: Standardized Z-score Chart for negative z-scores. < Area- Table IV Standard Normal Distribution .03 04 .05 z .00 .01 .02 .06 .07 .08 -3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002 -3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003 -3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005 -3.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007 -3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010 -2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014 -2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019 -2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026 -2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036 -2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048 -2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 -2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 -2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110 -2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143 -2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183 0.0262 0.0256 0.0250 0.0329 0.0064 0.0084 -1.9 0.0287 0.0244 0.0239 0.0281 0.0274 0.0268 0.0336 -1.8 0.0359 0.0351 0.0344 0.0322 0.0314 0.0307 0.0301 0.1190 0.1170 -1.0 0.1401 0.1379 0.1635 0.1611 -1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 -1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 -1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 -1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681 -1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 -1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985 -1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 -0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 -0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867 -0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148 -0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451 -0.5 0.3085 03050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776 -0.4 0.3446 03409 03372 0.3336 0.3300 0.3264 0.3228 03192 0.3156 0.3121 -0.3 0.3821 03783 03745 0.3707 0.3669 0.3632 0.3594 0.3557 03520 0.3483 -0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 03897 0.3859 -0.1 0.4602 0.4562 0,4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247 -0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4761 0.4721 0.4681 0.4641 0.4801 09 0.0233 0.0294 0.0367 0.0455 0.0559 Table C: T-distribution Chart t Table cum. prob one-tail two-tails df 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 t.75 t so 0.50 0.25 1.00 0.50 1.000 1.376 1.963 3.078 6.314 12.71 31.82 63.66 318.31 6.965 9.925 22.327 0.816 1.061 0.765 0.978 0.741 7.173 0.000 0.727 1.386 1.886 2.920 4.303 1.250 1.638 2.353 3.182 4.541 5.841 10.215 0.941 1.190 1.533 2.132 2.776 3.747 4.604 1.476 2.015 2.571 3.365 1.440 1.943 2.447 3.143 1.895 2.365 2.998 2.306 4.032 5.893 0.920 1.156 0.906 1.134 0.896 1.119 0.000 0.718 3.707 5.208 0.000 0.711 1.415 3.499 4.785 0.000 0.706 0.889 1.397 1.860 2.896 3.355 4.501 1.108 0.883 1.100 1.383 1.833 2.262 2.821 3.250 4.297 4.781 1.812 2.228 4.144 4.587 0.000 0.703 0.000 0.700 0.879 1.093 1.372 0.000 0.697 0.876 1.088 1.363 0.873 1.083 1.356 1.782 1.796 2.201 3.106 4.025 4.437 0.000 0.695 2.179 3.055 3.930 4.318 0.000 0.694 1.079 1.350 1.771 3.012 3.852 4.221 0.870 0.868 1.076 0.000 0.692 1.345 1.761 2.624 2.977 3.787 4.140 0.000 0.691 1.753 3.733 4.073 0.000 0.690 2.921 3.686 4.015 0.000 3.819 2.074 2.508 2.819 3.505 3.792 2.069 2.500 2.807 3.485 3.768 2.064 2.797 3.467 3.745 2.492 2.485 2.787 3.725 0.000 0.684 0.866 1.074 1.341 2.602 2.947 0.865 1.071 1.337 1.746 2.583 0.689 0.863 1.069 1.333 1.740 2.110 2.567 2.898 3.646 3.965 0.000 0.688 0.862 1.067 1.330 1.734 2.101 2.552 2.878 3.610 3.922 0.000 0.688 0.861 1.066 1.328 1.729 2.093 2.539 2.861 3.579 3.883 0.000 0.687 0.860 1.064 1.325 1.725 2.086 2.528 2.845 3.552 3.850 0.000 0.686 0.859 1.063 1.323 1.721 2.080 2.518 2.831 3.527 0.000 0.686 0.858 1.061 1.321 1.717 0.000 0.685 0.858 1.060 1.319 1.714 0.000 0.685 0.857 1.059 1.318 1.711 0.000 0.684 0.856 1.058 1.316 1.708 2.060 3.450 0.856 1.058 1.315 1.706 2.056 2.479 2.779 3.435 0.855 1.314 1.703 2.052 2.473 2.771 3.421 0.855 1.056 1.313 1.701 2.048 2.467 2.763 3.408 0.854 1.055 1.311 1.699 1.310 1.697 2.042 2.457 2.750 1.303 1.684 2.021 2.423 1.296 1.671 2.000 2.390 80 1.292 1.664 1.990 2.374 100 0.000 0.677 0.845 1.042 1.290 1.660 1.984 2.364 2.626 3.174 1000 0.000 1.037 1.282 1.646 1.962 2.330 2.581 3.098 1.036 1.282 1.645 1.960 2.326 2.576 3.090 3.291 80% 90% 95% 98% 99% 99.8% 99.9% Confidence Level 0.000 0.684 1.057 28 0.000 0.683 29 2.045 2.462 2.756 3.396 30 3.385 40 0.000 0.683 0.000 0.683 0.854 1.055 0.000 0.681 0.851 1.050 0.000 0.679 0.848 1.045 1.043 2.704 3.307 60 2.660 3.232 0.000 0.678 0.846 2.639 3.195 0.675 0.842 0.842 Z 60% 70% 26 27 0.000 0.000 0.000 0.000 tso tas 0.20 0.15 0.40 0.30 0.000 0.674 0% 50% t.90 0.10 0.20 t 95 0.05 0.10 t .975 0.025 0.05 t.99 0.01 0.02 t 995 0.005 0.01 t 999 0.001 0.002 2.764 3.169 2.718 2.681 2.160 2.650 2.145 2.131 2.120 t.9995 0.0005 0.001 636.62 31.599 12.924 8.610 6.869 5.959 5.408 5.041 3.707 3.690 3.674 3.659 3.646 3.551 3.460 3.416 3.390 3.300