Question

For a lottery to be​ successful, the public must have confidence in its fairness. One of the lotteries in a state is a​ pick-3 lottery, where 3 random digits are drawn each day. A fair game depends on every value​ (0 to​ 9) being equally likely at each of the three positions. If​ not, then someone detecting a pattern could take advantage of that and beat the lottery. To investigate the​ randomness, we'll look at the data collected over a​ 32-week period. Although the winning numbers look like​ three-digit numbers, in​ fact, each digit is a randomly drawn numeral. We have 654 random digits in all. Are each of the digits from 0 to 9 equally​ likely?

Group   Count   %
0   61   9.327
1   56   8.563
2   66   10.092
3   62   9.480
4   72   11.009
5   59   9.021
6   71   10.856
7   75   11.468
8   69   10.550
9   63   9.633

​d) Test an appropriate hypothesis and state your results.

Compute the appropriate test statistic.

The test statistic is ________

​(Round to three decimal places as​ needed.)

Please explain how you arrived at the answer. I would like to do it on my own with guidance.

Answer 1

Ho: likelihood of drawing of each numeral is equally likely

H1: likelihood of drawing of each numeral is not equally likely

chi square test of goodness of fit

here, n=654

each expected proportion = 1/10 = 0.1

expected frequency = np = 654*0.1 = 65.4

expected proprtion	observed,O		expected frequency ,E	(O-E)²/E
0.1	61		65.4	0.2960
0.1	56		65.4	1.3511
0.1	66		65.4	0.0055
0.1	62		65.4	0.1768
0.1	72		65.4	0.6661
0.1	59		65.4	0.6263
0.1	71		65.4	0.4795
0.1	75		65.4	1.4092
0.1	69		65.4	0.1982
0.1	63		65.4	0.0881
	654		654	5.2966

chi square stat=Σ(O-E)²/E = 5.297

DF=k-1 = 10- 1 = 9 [here, k is total category = 10 ]

p-value = 0.8077 [excel function: =chi.dist.rt(5.297, 9 ) ]

since, p-value >0.05, fail to reject Ho,

so, likelihood of drawing of each numeral is not equally likely

please revert for any doubts