5.5 Introduction to Hypothesis Tests
One job of a statistician is to make statistical inferences about populations based on samples taken from the population. Confidence intervals are one way to estimate a population parameter.
Another way to make a statistical inference is to make a decision about a parameter. For instance, a car dealership advertises that its new small truck gets 35 miles per gallon on average. A tutoring service claims that its method of tutoring helps 90% of its students get an A or a B. A company says that female managers in their company earn an average of $60,000 per year. A statistician may want to make a decision about or evaluate these claims. A hypothesis test can be used to do this.
A hypothesis test involves collecting data from a sample and evaluating the data. Then the statistician makes a decision as to whether or not there is sufficient evidence to reject the null hypothesis based upon analyses of the data.
In this section, you will conduct hypothesis tests on single means when the population standard deviation is known.
Hypothesis testing consists of two contradictory hypotheses or statements, a decision based on the data, and a conclusion. To perform a hypothesis test, a statistician will perform some variation of these steps:
- Define hypotheses.
- Collect and/or use the sample data to determine the correct distribution to use.
- Calculate test statistic.
- Make a decision.
- Write a conclusion.
Defining your hypotheses
The actual test begins by considering two hypotheses: the null hypothesis and the alternative hypothesis. These hypotheses contain opposing viewpoints.
The null hypothesis (H0) is often a statement of the accepted historical value or norm. This is your starting point that you must assume from the beginning in order to show an effect exists.
The alternative hypothesis (Ha) is a claim about the population that is contradictory to H0 and what we conclude when we reject H0.
Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.
After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are “reject H0” if the sample information favors the alternative hypothesis or “do not reject H0” or “decline to reject H0” if the sample information is insufficient to reject the null hypothesis.
The following table shows mathematical symbols used in H0 and Ha:
| H0 | Ha |
|---|---|
| Equal (=) | Not equal (≠) or greater than (>) or less than (<) |
| Equal (=) | Less than (<) |
| Equal (=) | More than (>) |
Figure 5.12: Null and alternative hypotheses
NOTE: H0 always has a symbol with an equal in it. Ha never has a symbol with an equal in it. The choice of symbol in the alternative hypothesis depends on the wording of the hypothesis test. Despite this, many researchers may use =, ≤, or ≥ in the null