Question

Mr. Brown the CEO of a corporate company wants to know if a new method of rewarding employees influences morale. He gives 12 employees an anonymous survey measuring morale, implements the new method, and then measures morale again after four weeks.

Employee	1	2	3	4	5	6	7	8	9	10	11	12
Before	4	7	1	6	6	3	5	6	6	2	8	7
After	6	6	5	7	8	6	9	7	8	4	9	9

a. State the null and alternative hypotheses

b. State the rejection region

c. Conduct a sign test on the above data

d. What can you conclude?

Answer 1

It is assumed that the values are independent and come from same continuous distribution.

a. Null hypothesis:

The Difference between the pairs(before and after) has zero median or the Differences are symmetric.

Alternative Hypothesis:

The Median is negative indicating that the after effect is more or the new method of rewarding employees influences morale.

To conduct the sign test we calculate the differences and count number of + and - signs.

Emp	before	after	Differences Sign
1	4	6	-2	-
2	7	6	1	+
3	1	5	-4	-
4	6	7	-1	-
5	6	8	-2	-
6	3	6	-3	-
7	5	9	-4	-
8	6	7	-1	-
9	6	8	-2	-
10	2	4	-2	-
11	8	9	-1	-
12	7	9	-2	-

Here Positive sign count = 1
Negative sign count = 11
Total count = 12

Test statistics:

We calculate Z-score : where X is the number of -ive signs and p is the null value of the proportion ie p=0.5(Indicating that the prob(+ive difference)=0.5.

In our case X=11, n=12 and p=0.5

Applying this we get,

  

  

The z-value is 2.88675. The p-value is .00195. The result is significant at p < 0.05. Therefore the Null hypothesis is rejected and hence the new method influences the employee morale.