CK-12 Probability and Statistics - Advanced

12.1. Introduction to Non-Parametric Statistics http://www.ck12.org

TABLE12.2:(continued)

Observation Number Referrals Before Program Referrals After Program

8 9 6

Since we need to determine the number of observations where there is a positive difference and the number of
observations where there is a negative difference, it is helpful to add an additional column to the table to classify
each observation as such (see below). We ignore all zero or equal observations.

TABLE12.3:

Observation Number Referrals Before Program Referrals After Program Change

1 8 5 −

2 10 8 −

3 2 3 +

4 4 1 −

5 6 4 −

6 4 1 −

7 5 7 +

8 9 6 −

When performing the sign test, we use thet-distribution if the sample has less than 30 observations and we use the
normal distribution if the sample has greater than 30 observations. Regardless of the distribution that we use, the
formula for calculating the test statistic (either thet- orz-score) is the same.

t=

|Positive Observations−Negative Observations|− 1

√

n

This formula states that the standard score (thezor thet) is equal to the absolute value of the difference between
positive differences within matched pairs and the negative differences within matched pairs minus one and divided
by the square root of the number of observations. For our example above, we would have a calculatedt-score of:

t=

| 2 − 6 |− 1

√

8

≈ 1. 06

Similar to other hypothesis tests using standard scores, we establish null and alternative hypotheses about the
population and use the test statistic to assess these hypotheses. As mentioned, this test is used with paired data
and examines whether the median of the two data sets are equal. When we conduct a pre-test and a post-test using
matched data, our null hypothesis is that the difference between the data sets will be zero. In other words, under
our null hypothesis we would expect there to be some fluctuations between the pre- and post-tests, but nothing of
significance.

H 0 :m= 0

Ha:m 6 = 0

With the sign test, we set criterion for rejecting the null hypothesis in the same way as we did when we were testing
hypotheses using parametric tests. For the example above, if we setα=.05 we would have critical values set at
2 .37 standard scores above and below the mean. Since our standard score of 1.06 does not exceed the critical value