**Question 21**

The following arrangement of men, M, and women, W, lined up to purchase tickets for a rock concert:

Test for randomness at the

**α**= 0.05 significance level

**Solution Steps**

In this case, we see that the sample size is fairly large, so we are going to use a slightly different method in this case. We are going to calculate the mean runs and the standard deviation.

Then we go ahead to find the z-statistic. We would use the following 7 steps to disolve this problem.

Step 1: State the null and alternate hypothesis

Step 2: Determine the number of runs

Step 3: Calculate the mean runs

Step 4: Calculate the Standard deviation

Step 5: Calculate the z-Statistic

Step 6: Determine the Critical value

Step 7: Draw a conclusion

### Step 1: State the null and alternate hypothesis

H

_{0}: Arrangement is random

H

_{1}: Arrangment is not random

### Step 2: Determine the Number of Runs

This means that you need to mark each of the categories so you can easily count them

Here I mark each run alternatively with red for M and black wor W. The outcome is given below.

The number of runs is given by R = 27

The number of Men, n

_{1}= 30

The number of Women, n

_{2}= 18

### Step 3: Calculate the Mean runs

The mean is given by the formula

### Step 4: Calculate the Standard Deviation

You can find the standard deviation for a runs test using the formula

We can go ahead to substitute the value of n

_{1}= 30 and n

_{2}= 18

### Step 5: Calculate the z-Statistic

The z-Statistic can be calculated using the formular

Then we can substitute the value of

R = 27

μ

_{R}= 23.5

σ

_{R}= 3.2083

### Step 6: Determine the Critical Value

Look up the value of the critical value from statistical table of normal distribution

We get a critical value for of

**1.96**

### Step 7: State the Decision

Since the calculated value of z = 1.0909 is within the accept region(less than the critical value of 1.96), we therefore accept (fail to reject) the null hypothesis and conclude that there is not real evidence that the arrangement is not random

Leave a comment if you have any observation or difficulty following this lesson

Thank you for solving.