12.1

7. The first significant digit in any number must be 1, 2, 3, 4, 5, 6, 7, 8, or 9. It was discovered that first digits do not occur with equal frequency. Probabilities of occurrence to the first digit in a number are shown in the accompanying table. The probability distribution is now known as Benford's Law. For example, the following distribution represents the first digits in 201 allegedly fraudulent checks written to a bogus company by an employee attempting to embezzle funds from his employer. Complete parts (a) and (b) below. Click here to view the tables." Click here to view the table of critical values of the chi-square distribution. 5 (a) Because these data are meant to prove that someone is guilty of fraud, what would be an appropriate level of significance when performing a goodness-of-fit test? Use a = 0.01. (b) Using the level of significance chosen in part (a), test whether the first digits in the allegedly fraudulent checks obey Benford's Law. Do the first digits obey Benford's Law? What are the null and alternative hypotheses? A. Ho: The distribution of the first digits in the allegedly fraudulent checks obeys Benford's Law. Hy: The distribution of the first digits in the allegedly fraudulent checks does not obey Benford's Law. O B. Ho: The distribution of the first digits in the allegedly fraudulent checks does not obey Benford's Law. H,: The distribution of the first digits in the allegedly fraudulent checks obeys Benford's Law. What is the test statistic? (Round to three decimal places as needed.) What is the range of P-values for the test? The range of P-values for the test is (1) Using the P-value approach, compare the range of P-values with the given a = 0.01 level of significance. Based on the results, do the first digits obey Benford's Law? O A. Reject the Ho because the range of P-values is less than the given a level of significance. O B. Do not reject the Ho because the range of P-values is greater than the given a level of significance O c. Reject the Ho because the range of P-values is greater than the given a level of significance O D. Do not reject the Ho because the range of P-values is less than the given a level of significance. (c) Based on the results of part (b), could one think that the employee is guilty of embezzlement? O A. Yes, the first digits do not obey Benford's Law. O B. No, the first digits obey Benford's Law. O C. No, the first digits do not obey Benford's Law. O D. Yes, the first digits obey Benford's Law. 4: Distribution of First Digits Full data set Distribution of first digits (Benford's Law) Digit 2 Probability 0.301 0.176 0.125 .097 0.079 Digit 6 7 8 9 Probability .067 0.058 0.051 0.046 First digits in allegedly fraudulent checks First digit 1 2 3 4 5 6 9 Frequency 42 25 28 26 24 17 15 1 7 6: Chi-Square Distribution Critical Values Table Chi-Square (x ) Distribution Degrees of Areato the Right of Critical Value Freedom 0.995 0.99 0.975 0.95 0.90 0.10 0.05 0.025 0.01 0.005 0.001 0.004 0.016 2.706 3.841 5.024 6.635 1879 0.010 0 020 0.061 0.10 4.605 3091 7.378 9.210 10 *07 0.072 0.216 0.352 0.584 6:251 7815 9 348 11.345 12838 0.207 0297 0.484 0.711 1524 7.779 9488 11.143 13.277 14.860 0.412 0554 0.831 1.145 9.236 11.070 12.833 5.086 16.750 0.676 0.872 1.635 10.64 12592 14.449 16.812 18.54 0.989 1.239 1.690 2.16 2.833 12.01 4.067 16.013 8.475 2027. 1.344 1.646 2.180 2.73 3.490 13.36. 15.50 17535 20.09 21.95 1.735 2.088 2.70 3.32 4.168 14.68 16.919 19.02 21.66 2358 2.156 2.538 3.247 3.940 4.865 15.987 18.30 20.48 23 209 2518 2603 3.063 3.816 4.575 5.578 17275 19.67 21.920 24.72 26.75 3074 3 571 4.40 5.226 6.30 18.549 21.02 23.33 26 217 28.300 4 107 5 009 5 892 7042 19.81 22.362 24 736 29 81 4 075 4.660 6.571 7.790 21.064 23.685 26.119 29141 31.319 4601 5.229 5.262 7261 8.547 22.307 24.996 27.488 30.578 32.801 $142 5.812 6.908 2962 9.312 23.54 26.296 28.84 3200 34.267 5.697 6.408 7564 8.672 10.085 27587 30.191 33.409 35.718 6.265 7015 8.231 9.390 25.989 31.52 34.805 37156 6.84 7633 8.907 10.117 11.651 27204 30.144 32.85 36.191 38.582 7434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37566 39.997 8.034 8.897 10.283 11.591 13.240 29.61 32.671 35.479 38.932 41.401 8.643 9.542 10.982 12.338 14.041 30.81 33.924 36.78 40.28 42.796 9.260 10.196 11.689 13.091 14.848 32.00 35.172 38.076 41638 44.181 9.886 10 856 12.401 13.848 15.659 33.195 36.415 39.36 42980 45.559 10.520 1.524 13.120 14.611 16.473 34.38 37652 40.646 44 314 46.928 11.160 12198 13.844 15.379 17.292 35.56 38,885 41.923 45.642 48.290 11.808 12879 14.573 16.151 18.114 36.741 40.113 43.195 46 963 12461 49 845 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50993 13.121 14.256 16.047 17.708 39.087 42557 45.72 49.58 52.336 13.78 14.953 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53672 20.707 22.164 24.433 26.509 29.051 $1.80 35.758 59.34 53.691 66.766 2799 9.707 32.357 34.764 63.16 67505 71.420 76.154 79 490 35 53 37.48 40.482 43.188 46.459 74.39 79.082 83.29. 88.379 91.952 43 275 45.442 48.758 $1.739 55.329 85.527 90.531 95.023 100.425 104 215 $1.17 53.540 57153 60.391 64.278 96.578 101.879 105.629 112.329 116.321 100 5919 61.75 65.647 69.126 73.291 107.565 113.145 118.136 124.116 128.299 67328 70.065 74.222 77929 82.358 118.498 124.342 129.561 135 807 140.169 Left tail Right tail Area = 1- Two tails X2 - - The area to the Xi- The area to the The area to the right right of this right of this of this value is . value is av. value is 1- a. The area to the right of this value is1 - -. (1) O between 0.05 and 0.10. O between 0.025 and 0.05. O less than 0.01. O between 0.01 and 0.025