Question

Using the Runs Test for Randomness. In Exercises 5–10, use the runs test with a significance level of\(\alpha \)= 0.05. (All data are listed in order by row.)

Draft Lottery In 1970, a lottery was used to determine who would be drafted into the U.S. Army. The 366 dates in the year were placed in individual capsules, they were mixed, and then capsules were selected to identify birth dates of men to be drafted first. The first 30 results are listed below. Test for randomness before and after the middle of the year, which is July 1.

Sept.14	Apr.24	Dec.30	Feb.14	Oct.18	Sept.6	Oct.26	Sept.7	Nov.22
Dec.6	Aug.31	Dec.7	July 8	Apr.11	July12	Dec.29	Jan.15	Sept.26
Nov.1	June 4	Aug.10	June26	July24	Oct.5	Feb.19	Dec.14	July 21
June 5	Mar.2	Mar.31

Short answer

It can be concluded that the given sample of dates is random.

Step-by-step solution

Given information

The question provides data on 30 randomly selected army men showing their birth dates.

Identify the hypothesis

The researcher wants to test the claim that the sequence of dates israndomly selected before and after July 1.

The null hypothesis is as follows:

The given sequence of dates is random.

The alternative hypothesis is as follows:

The given sequence of dates is not random.

Data arrangement

Identify the dates before July 1 with the symbol B.

Identify the dates after July 1 with the symbol A.

The table below shows the indicated symbols:

Sept.14	A
Apr.24	B
Dec.30	A
Feb.14	B
Oct.18	A
Sept.6	A
Oct.26	A
Sept.7	A
Nov.22	A
Dec.6	A
Aug.31	A
Dec.7	A
July 8	A
Apr.11	B
July 12	A
Dec.29	A
Jan.15	B
Sept.26	A
Nov.1	A
June 4	B
Aug.10	A
June 26	B
July 24	A
Oct.5	A
Feb.19	B
Dec.14	A
July 21	A
June 5	B
Mar.2	B
Mar.31	B

Step 4:Calculate the test statistic

The sequence is as follows:

A

B

A

B

A

A

A

A

A

A

A

A

A

B

A

A

B

A

A

B

A

B

A

A

B

A

A

B

B

B

Now, the number of times A occurs is denoted by\({n_1}\), and the number of times B occurs is denoted by\({n_2}\)

.

Thus,

\(\begin{array}{l}{n_1} = 20\\{n_2} = 10\end{array}\)

The runs of the sequence are formed as follows:

\(\underbrace A_{{1^{st}}run}\underbrace B_{{2^{nd}}run}\underbrace A_{{3^{rd}}run}\underbrace B_{{4^{th}}run}\underbrace {AAAAAAAAA}_{{5^{th}}run}\underbrace B_{{6^{th}}run}\underbrace {AA}_{{7^{th}}run}\underbrace B_{{8^{th}}run}\underbrace {AA}_{{9^{th}}run}\underbrace B_{{{10}^{th}}run}\underbrace A_{{{11}^{th}}run}\underbrace B_{{{12}^{th}}run}\underbrace {AA}_{{{13}^{th}}run}\underbrace B_{{{14}^{th}}run}\underbrace {AA}_{{{15}^{th}}run}\underbrace {BBB}_{{{16}^{th}}run}\)

The number of runs denoted by G is equal to 16.

Since \({n_1} \le 20\) and \({n_2} \le 20\), the test statistic is denoted by G and has a value equal to 16.

Determine the critical value and the conclusion of the test

The critical values of G for\({n_1} = 20\)and\({n_2} = 10\)are 9 and 20.

Since the value of G equal to 16 lies between the critical values, so the decision is fail to reject the null hypothesis.

There is not enough evidence to conclude that the given sample is not random.

In other words, there is not enough evidence to warrant rejection of the claim that the sequence of dates is randomly selected before and after July 1.

Thus, it can be said that the sequence of dates is random.