For a lottery to be successful, the public must have condence in its fairness. One of the lotteries in Maryland is 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 data collected over a recent 32-week period. Althought the winning numbers look like three digit numbers, in fact, each digit is a randomly drawn 1 numeral. We have 654 random digits in all. Are each of the digits from 0 to 9 equally likely? Here is a table of the frequencies.

Group Count %
0 62 9.480
1 55 8.410
2 66 10.092
3 64 9.786
4 75 11.468
5 57 8.716
6 71 10.856
7 74 11.315
8 69 10.550
9 61 9.327

(a) Set up the relevant null and alternative hypothesis.
(b) prepare down the test statistic to be used for testing the hypothesis.
(c) What is the sampling distribution of the test statistic if the null hypothesis is true?
(d) If the level of signicance  = 5%, prepare down the rejection region.
(e) Based on the data, do you reject the null hypothesis? Justify.
(f) Did you commit a type I error based on your conclusion to 2e above?