Significance Testing
Prerequisites
Binomial
Distribution,
Introduction to Hypothesis Testing
Learning Objectives
 Describe how a probability value is used to cast doubt on the null
hypothesis
 Define "statistically significant"
 Distinguish between statistical significance and practical significance
 Distinguish between two approaches to significance testing
A low probability
value casts doubt on the null hypothesis. How
low must the probability value be in order to conclude that the
null hypothesis is false. Although there is clearly no right or
wrong answer to this question, it is conventional to conclude
the null hypothesis is false if the probability value is less
than 0.05. More conservative researchers conclude the null hypothesis
is false only if the probability value is less than 0.01. When
a researcher concludes that the null hypothesis is false, the
researcher is said to have rejected the null hypothesis. The probability
value below which the null hypothesis is rejected is called the
α level or simply α. It is also
called the significance level.
When the null hypothesis is rejected, the effect
is said to be statistically significant.
It is very important to keep in mind that statistical significance
means only that the null hypothesis of exactly no effect is rejected;
it does not mean that the effect is important, which is what "significant"
usually means. When an effect is significant, you can have confidence
the effect is not exactly zero. Finding that an effect is significant
does not tell you about how large or important the effect is.
Do not confuse statistical significance with
practical significance. A small effect can be highly significant
if the sample size is large enough.
There are two approaches (at least) to conducting
significance tests. In one (favored by R. Fisher) a significance
test is conducted and the probability value reflects the strength
of the evidence against the null hypothesis.
The alternative approach (favored by the statisticians
Neyman and Pearson) is to specify an α
level before analyzing the data. If
the data analysis results in a probability value below the α
level, then the null hypothesis is rejected; if it is not, then
the null hypothesis is not rejected. According to this perspective,
if a result is significant, then it does not matter how significant
it is. Moreover, if it is not significant, then it does not matter
how close to being significant it is.
The former approach (preferred by Fisher) is more
suitable for scientific research and will be adopted here. The
latter is more suitable for applications in which a yes/no decision
must be made.
