Using Nonparametric Tests. In Exercises 1–10, use a 0.05 significance level with the indicated test. If no particular test is specified, use the appropriate nonparametric test from this chapter.

California Lottery Listed below are consecutive first-digits drawn in the California Daily 4 lottery. Test for randomness of even and odd integers. Does the lottery appear to be working as it should?

5	2	2	8	4	8	8	7	1	0	6	4	1	5	1	5	5
3	1	4	1	5	0	0	3	9	6	6	3	7

The first digits drawn in a lottery are given.

This test is used to investigate the randomness of a sequence.

The null hypothesis is as follows:

The sequence of odd and even digits is random.

The alternative hypothesis is as follows:

The sequence of odd and even digits is not random.

The data is arranged in the following manner, symbolizingall odd digits with the letter O and all even digits with the letter E:

The sequence is as follows:

Now, the number of times E occurs is denoted by\({n_1}\)and the number of times O occurs is denoted by\({n_2}\).

Thus,

\(\begin{array}{l}{n_1} = 14\\{n_2} = 16\end{array}\)

The runs of the sequence are formed as follows:

\(\underbrace O_{{1^{st}}run}\underbrace {EEEEEE}_{{2^{nd}}run}\underbrace {OO}_{{3^{rd}}run}\underbrace {EEE}_{{4^{th}}run}\underbrace {OOOOOOO}_{{5^{th}}run}\underbrace E_{{6^{th}}run}\underbrace {OO}_{{7^{th}}run}\underbrace {EE}_{{8^{th}}run}\underbrace {OO}_{{9^{th}}run}\underbrace {EE}_{{{10}^{th}}run}\underbrace {OO}_{{{11}^{th}}run}\)

The number of runs is denoted by G is equal to 11.

Here, \({n_1} < 20\) and \({n_2} < 20\). The value of the test statistic is G.

Referring to Table A-10, the critical values of G at\(\alpha = 0.05\)for\({n_1} = 14\)and\({n_2} = 16\)are 10 and 22.

The value of G falls within the interval (10, 22).

Thus, the null hypothesis is failed to reject.

There is not enough evidence to warrant rejection of the claim that the sequence of digits (odd and even) is random.

Thus, the sequence of the first digits is random.

Since the sequence of digits occurs randomly, it can be said that the lottery is fair and is working as it should.