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. (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. H,: 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? XO = (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. 1: Distribution of First Digits Full data set Distribution of first digits (Benford's Law) Digit 5 Probability 0.301 0.176 0.125 ).097 0.079 Digit 9 Probability 0.067 0.058 0.051 0.046 First digits in allegedly fraudulent checks First digit 1 2 3 4 5 6 7 8 9 Frequency 42 25 28 26 24 17 15 17 7 2: Chi-Square Distribution Critical Values Table Chi-Square (x ) Distribution Degrees of Area to 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 7.879 0.010 0 020 0.061 0.10 0.211 4.605 5991 7.378 9.210 10.597 0.072 0115 0.216 0.352 0.584 6.251 7815 9.348 11.34 12838 0.207 0297 0.484 D.71 1 1064 7.779 9488 11.143 13.277 14.860 0.412 0554 0.831 1.145 1.610 9.236 11.070 12.833 15.086 16.750 0.676 0.872 1.237 1.635 2.204 10.64 12.59 14.449 16.812 18.54 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 1.344 1.646 2.180 2.733 20218 3.490 13.362 3.507 17535 20.090 21.955 1.735 2.088 2.70 3.325 4.168 14.68 16.919 19.02 21.66 23589 2.156 2.538 3.247 3.940 4.865 15.987 18.307 20.483 23 209 25.188 2603 3.063 3.816 4.575 5.578 17275 19.67 21.92 24.72 26 757 3074 3.571 4.40 5.226 6.304 18.549 21.02 23.33 26 217 28 300 3565 4.107 5.009 5.892 7042 19.812 24.736 27688 29 819 4075 4.660 6.571 7.790 23.685 26.119 29 141 31.319 4601 5.229 6.262 7261 8.547 22.307 24.996 27.488 30578 32.801 $142 5.812 6.908 7962 9.312 23.542 26.296 28.845 32000 34.267 5.697 6.408 7564 8.672 10.085 24.769 27587 30.19 33.409 35.718 6.265 7015 8.231 9.390 10.865 25.989 28.86 31.52 34.80 37156 6.844 7633 8.907 10.117 11.651 27204 30.14 32.85 36.191 38.582 2434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37566 39.997 8.034 8.807 10.283 11.591 13.240 29.61 32.67 35.47 38 93 41.401 8.643 9.542 10.982 12.338 14.041 30.813 33.92 36.78 40 28 42.796 9.260 10.196 11.689 13.091 14.848 32.007 35172 38.076 41 638 44.181 9.886 10 856 12.401 13.848 15.659 33.19 36.415 39 364 42.980 45.559 10.520 11.524 13.120 14.611 16.473 34.382 37652 40.64 44.314 46928 11.160 2198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290 11808 12879 14573 16.151 18.114 36.741 40.11 43.19 46.96 49 645 12461 13.565 15.308 16.928 18.939 37.916 44 461 48.278 50 993 13.12 14.256 17.708 19.768 39.087 42557 49.588 52 3 36 13.78 14.953 16.791 18.49 20.599 40.256 43.773 46.979 50.892 53 672 20.707 22.164 24.433 26.509 29.051 59.34 63 691 66.766 2799 29.707 32.357 34.764 37689 51.805 63.16 67505 71.420 79 490 3553 37.485 40.482 43.188 16.459 74.397 83.29. 88.379 91.952 4327 45.442 48.758 51.739 $5.329 85.52 90.531 95.023 100.425 104 215 $1.17 53.540 57153 60.391 64.278 96.578 101.879 106.629 112.329 116.321 5919 61.75 65.647 69.126 73.20 107.565 113.145 118.136 124116 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 x: - The area to the xi-. - The area to the xi-s . The area to the right right of this right of this of this value is a . value is a. value is 1-a. The area to the right of this value is 1 - (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. greater than 0.10
