#### Subject 2. Null Hypothesis and Alternative Hypothesis

The null hypothesis (designated H0) is the statement that is to be tested. The null hypothesis is a statement about the value of a population. The null hypothesis will either be rejected or fail to be rejected.

• For example, the null hypothesis could be: "The mean monthly return for stocks listed on the Vancouver Stock Exchange is not significantly different from 1%." Note that this is the same as saying the mean (μ) monthly return on stocks listed on the Vancouver Stock Exchange is equal to 1%. This null hypothesis, H0, would be written as: H0: μ = 1%.

• As another example, if a null hypothesis is stated as "There is no difference in the revenue growth rate for satellite TV dishes before and after the negative TV advertising campaign aired by the cable industry," then the null hypothesis could be written to show that two rates are equal: H0: r1 = r2.

It is important to point out that accepting the null hypothesis does not prove that it is true. It simply means that there is not sufficient evidence to reject it.

Note that it makes no sense to hypothesize about known sample values, for the simple reason that they are known, just like it makes no sense to construct confidence intervals or obtain point estimates for known values. Hypothesis tests are carried out on unknown population parameters.

The alternate hypothesis is the statement that is accepted if the sample data provides sufficient evidence that the null hypothesis is false. It is designated as H1 and is accepted if the sample data provides sufficient statistical evidence that H0 is false.

The following example clarifies the difference between the two hypotheses. Suppose the mean time to market for a new pharmaceutical drug is thought to be 3.9 years. The null hypothesis represents the current or reported condition and would therefore be H0: μ = 3.9. The alternate hypothesis is that this statement is not true, that is, H1: μ ≠ 3.9. The null and alternative hypotheses account for all possible values of the population parameter.

There are three basic ways of formulating the null hypothesis.

• H0: μ = μ0 versus H1: μ ≠ μ0. This hypothesis is two-tailed, which means that you are testing evidence that the actual parameter may be statistically greater or less than the hypothesized value.

• H0: ≤ μ0 versus H1: μ > μ0. This hypothesis is one-tailed; it tests whether there is evidence that the actual parameter is significantly greater than the hypothesized value. If there is, the null hypothesis is rejected. If there is not, the null hypothesis is accepted.

• H0: μ ≥ μ0 versus H1: μ < μ0. This hypothesis is one-tailed; it tests whether there is evidence that the actual parameter is significantly less than the hypothesized value. If there is, the null hypothesis is rejected. If there is not, the null hypothesis is accepted.

The question most likely to be raised at this point is how do you know if a test is one-sided or two-sided? The general rule is as follows:

• If a question makes it clear that only one direction is to be examined, use a one-sided test.
• If there is no clue in the question as to which direction should be examined, use a two-sided test.

Normally, there is little ambiguity; the question will make it clear which test should be used. A question often asked involves testing whether a population mean is greater than or less than a specific number. In this case, use a one-tailed test. If the question asks you to test whether a population mean is different from a specific number, use a two-tailed test.

With practice, you'll see that this issue is not really a huge problem.