BBy Bot
Jun 09'24

Exercise

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.

Distribution of choices in the Powerball lottery.

Do you think that people are choosing numbers randomly? Justify your answer. The goal of this problem is to check the hypothesis that the chosen numbers are uniformly distributed. To do this, compute the value [math]v[/math] of the random variable [math]\chi^2[/math] given in Example. In the present case, this random variable has 44 degrees of freedom. One can find, in a [math]\chi^2[/math] table, the value [math]v_0 = 59.43[/math] , which represents a number with the property that a [math]\chi^2[/math]-distributed random variable takes on values that exceed [math]v_0[/math] only 5% of the time. Does your computed value of [math]v[/math] exceed [math]v_0[/math]? If so, you should reject the hypothesis that the contestants' choices are uniformly distributed.

Numbers chosen by contestants in the Powerball lottery.
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