The Null and Alternative
The most common problem I noticed on this assignment was caused by failing to properly identify the appropriate null and alternative hypotheses. In part, this is due to the Evans text’s somewhat confusing explanation of how to do this – the “burden of proof” approach. There is a much simpler approach that always works. First, make sure you closely read the problem statement looking for key words and phrases. This table may help:
The null hypothesis always is “no effect” or “no difference.” The null says that nothing new happened or the groups have the same mean – the status quo. The null states that a population parameter is equal to some value. It always is written with an equality statement, i.e. =, ≥, ≤. We always assume the null to be true.
The alternative states that the population parameter is different than the value in the null. It often is what is believed to be true or what we hope to prove. The alternative is the complement of the null and always contains an inequality statement, i.e. ≠, <, >. Initially, the alternative is assumed to be false, but it might be accepted if sufficient evidence is obtained.
Next, decide which of the hypotheses represents the claim. The claim can be either the null or the alternative. This is where the “burden of proof” framework is suggested. The rational is based on the underlying principle of hypothesis tests that the null is always assumed to be true but cannot be proven to be true. Based upon the evidence obtained, we can statistically “reject” the null if we have sufficient evidence supporting the alternative. But, because of the possibility of having sampling errors (random occurrences) in our data, we can never prove the null is true, only that we failed to get sufficient evidence to support the alternative; thus we “fail to reject” the null.
On the other hand, we can hope to gather sufficient evidence to support the alternative hypothesis. If we do, we can then “reject the null.” If the null is rejected, the alternative is accepted as being true.
Under the burden of proof conceptualization, if we want to prove the claim, we have to make it the alternative. However, this approach becomes problematic if the claim in the problem statement contains an equality, as this means the claim must be the null. Forcing the claim containing an equality to be the alternative forces us to change the actual wording of the problem statement. And that can lead to confusion and mistakes in setting up and running the actual hypothesis test.
My suggested approach is to always set the null and alternative first, and then decide which one is the claim. Although we cannot prove the claim to be true if it is the null, we can get evidence which will allow us to “fail to reject” it. In a practical (though not technical) sense, this is effectively the same thing.
Your test can be left/lower-tailed, right/upper-tailed, or two-tailed. The direction is determined by the alternative hypothesis. If the mathematical statement of Ha contains <, the symbol is pointing to the left and you have a left-tailed test. Same thought process for Ha containing >. If Ha contains ≠, you have a two-tailed test.
