There are several aspects of our text's description of hypothesis testing that are confusing and perhaps a bit misleading. The text advises you to read the word question, identify where the equivalence sign is, and set up our null and alternative hypotheses accordingly. Depending on how the problem is stated, this may or may not be the correct way to proceed. It is much better (and more correct) to set up the null and alternative hypotheses based on the scientific question at hand and then insert the equivalence sign depending on whether you wish to support or not support the claim.

Remember the following:

If you wish to support a claim, place it in the alternative hypothesis.
If you wish to not support a claim, place it in the null hypothesis.

The null hypothesis always contains the equivalence sign and we never try to prove that the null hypothesis is true. Instead, we start with the assumption that the null is true, and look for evidence to show that it is not. If we find sufficient evidence, we reject the null and conclude that the alternative is probably true. If we don't find sufficient evidence, we do not accept the alternative -- we just fail to reject the null. This is a very important point! We never try to prove anything with statistics. Instead, we establish a null hypothesis and look for sample-based evidence to try and reject the null. If we can't reject the null, we do not accept the alternative, we reserve judgement and state that there is insufficient evidence to conclude that the alternative is true.

In other words, the null hypothesis is what we believe to be true, in the absence of compelling evidence to the contrary. It is the hypothesis to be tested. If we reject the null, we can say that the alternative is probably true. If we fail to reject the null, we are not confident that the alternative is true--we merely do not have enough information to say that the null is untrue.

Supporting a claim is synonymous with finding enough evidence to conclude with confidence that the claim is true. Again, we do not look for evidence that the null is true. We look for evidence that the null is false, in order to say that the alternative is true. Therefore, in order to support a claim, we must set the claim up as the alternative hypothesis.

The text sometimes asks us to "test" claims without specifying whether we should be looking to support or not support the claim. This can easily lead to confusion and a degree of arbitrariness which hinges on whether or not we believe the source of the claim. Because this can lead to problems, it is better to have a clear statement based on the scientific question at hand and then insert the equivalence sign depending on whether you wish to support or not support the claim.

Consider a problem involving the mean cost of automobile repairs.

Suppose the claim is that the mean repair cost is less than $100 (u < 100). Since you work for the repair shop it is very logical that you wish to support the claim which is in the alternative hypothesis. Thus, we have:

Ho: u >= 100
Ha: u < 100 (claim)

Our sample statistics are: n = 5, x-bar = 75, s = 12.50 , alpha = .01, population is assumed to be normal.

The test statistic is computed as: t = -4.472 The critical value from the t-table is: t = -3.747 Our decision is to reject the Ho and conclude that the alternative is probably true. Hence, it is very likely that the mean repair bill is less than $100 and we support the claim.

Now, instead of the above, suppose the repair shop's claim was that the mean repair cost was at least $50. In this case, we place the claim in the null hypothesis and write:

Ho: u >= 50 (claim)
Ha: u < 50

Assuming that the sample statistics are the same as stated above, it is not necessary to perform a hypothesis test since x-bar is >= $50. This is true because the test statistic is positive (+4.472) and the critical t-value is negative (-3.747). Thus, there is no way we can reject the Ho. This example clearly demonstrates why it is imperative that we establish our hypotheses prior to taking our sample. We could easily be interested in either of the two null hypotheses so we need to identify the hypothesis of interest before we take our sample and draw our inference. Our conclusion will differ depending on our stated hypothesis.

Suppose you wish to refute the claim that the mean repair bill is less than or equal to $100. For this case our hypotheses are:

Ho: u <= 100 (claim)
Ha: u > 100

In this case, if your sample-based evidence leads you to reject the null hypothesis you can conclude that the alternative is probably true. This means that the mean repair cost is greater than $100 and you have refuted your claim. If you fail to reject the null hypothesis, you reserve judgment and do not conclude that the mean repair cost is <= $100.