Reference no: EM13970219

1. (a) Compute the leading digits of the first 100 powers of 2, and see how well these data fit the Benford distribution.

(b) Multiply each number in the data set of part (a) by 3, and compare the distribution of the leading digits with the Benford distribution.

2. In the Powerball lottery, contestants pick 5 different integers between 1 and 45, and in addition, pick a bonus integer from the same range (the bonus integer can equal one of the first five integers chosen). Some contestants choose the numbers themselves, and others let the computer choose the numbers. The data shown in Table are the contestant-chosen numbers in a certain state on May 3, 1996. A spike graph of the data is shown in Figure.

The goal of this problem is to check the hypothesis that the chosen numbers are uniformly distributed. To do this, compute the value v of the random variable χ2 given in Example 5.6. In the present case, this random variable has 44 degrees of freedom. One can find, in a χ2 table, the value v0 = 59.43 , which represents a number with the property that a χ2-distributed random variable takes on values that exceedv0 only 5% of the time. Does your computed value of v exceed v0? If so, you should reject the hypothesis that the contestants' choices are uniformly distributed.

Integer	Times Chosen	Integer	Times Chosen	Integer	Times Chosen
1	2646	2	2934	3	3352
4	3000	5	3357	6	2892
7	3657	8	3025	9	3362
10	2985	11	3138	12	3043
13	2690	14	2423	15	2556
16	2456	17	2479	18	2276
19	2304	20	1971	21	2543
22	2678	23	2729	24	2414
25	2616	26	2426	27	2381
28	2059	29	2039	30	2298
31	2081	32	1508	33	1887
34	1463	35	1594	36	1354
37	1049	38	1165	39	1248
40	1493	41	1322	42	1423
43	1207	44	1259	45	1224

Table: Numbers chosen by contestants in the Powerball  lottery.