removed from expected as we got. A decision about significance can then be made by comparing the
obtained P -value with a stated value of α.
example: Suppose it turns out that Todd’s 50 throws are approximately normally distributed with
mean 47.5 yards and standard deviation 8 yards. His claim is that he can average 50 yards per
throw. What is the probability of getting an observed mean this far below the expected 50 yards
by chance alone (that is, what is the P -value) if his true average is 50 yards (assume the
population standard deviation is 8 yards)? Is this finding significant at α = 0.05? At α = 0.01?
solution: We are assuming the population is normally distributed with mean 50 and standard
deviation 8. The situation is pictured below:This is the P -value: it’s the probability of getting a sample mean as far below 50 as we did by chance
alone, assuming the model based on a mean of 50 yards is correct. This finding is significant at the 0.05
level but not (quite) at the 0.01 level.
The Hypothesis-Testing Procedure
So far we have used confidence intervals to estimate the value of a population parameter (μ, p, μ 1 – μ 2 ,
p 1 – p 2 ). In the coming chapters, we test whether the parameter has a particular value or not. More
accurately, might ask if we have convincing evidence against the hypothesis that p 1 – p 2 = 0 or if μ = 3,
for example. That is, we will test the hypothesis that, say, p 1 – p 2 = 0. In the hypothesis-testing
procedure, a researcher does not look for evidence to support this hypothesis, but instead looks for
evidence against the hypothesis. The process looks like this.
• State the null and alternative hypotheses in the context of the problem . The first hypothesis, the null
hypothesis , is the hypothesis we are actually testing. The null hypothesis usually states that there is
nothing going on: the claim is correct or that there is no distinction between groups. It is symbolized by
H 0 . An example of a typical null hypothesis would be H 0 : μ 1 – μ 2 = 0 or H 0 : μ 1 = μ 2 . This is the
hypothesis that μ 1 and μ 2 are the same, or that populations 1 and 2 have the same mean. Note that μ (^1)
and μ 2 must be identified in context (for example, μ 1 = the mean score for all people in the population
before training).
The second hypothesis, the alternative hypothesis , is the theory that the researcher wants to