http://www.ck12.org Chapter 12. Non-Parametric Statistics
of 2.37, we would fail to reject the null hypothesis and cannot conclude that there is a significant difference between
the pre- and the post-test scores.
Using the Sign Test to Evaluate a Hypothesis about a Median of a Population
In addition to using the sign test to calculate standard scores and evaluate a hypothesis, we can also use it as a quick
and dirty way to estimate the probability of obtaining a certain number of successes or positives if there was no
difference between the observations in the matched data set. When we use the sign test to evaluate a hypothesis
about a median of a population, we are estimating the likelihood or theprobabilitythat the number of successes
would occur by chance if there was no difference between pre- and post-test data. Therefore, we can test these types
of hypotheses using the sign test by either (1) conducting an exact test using the binomial distribution when working
with small samples or (2) calculating a test statistic when working with larger samples as demonstrated in the section
above.
When working with small samples, the sign test is actually the binomial test with the null hypothesis that the
proportion of successes will equal 0.5. So how do these tests differ? While we use the same formula to calculate
probabilities, the sign test is a specific type of test that has its own tables and formulas. These tools apply only to the
case where the null hypothesis that the proportion of successes will equal 0.5 and not to the more general binomial
test.
As a reminder, the formula for the binomial distribution is:
P(r) =N!
r!(N−r)!pr( 1 −p)N−rwhere:
P(r) =the probability of exactly r successes
N=the number of observations
p=the probability of success on one trialSay that a physical education teacher is interested on the effect of a certain weight training program on students’
strength. She measures the number of times students are able to lift a dumbbell of a certain weight before the
program and then again after the program. Below are her results:
TABLE12.4:
Before Program After Program Change
12 21 +
9 16 +
11 14 +
21 36 +
17 28 +
22 20 −
18 29 +
11 22 +If the program had no effect, then the proportion of students with increased strength would equal 0.5. Looking at
the data above, we see that 6 of the 8 students had increased strength after the program. But is this statistically
significant? To answer this question we use the binomial formula:
P(r) =N!
r!(N−r)!
pr( 1 −p)N−r