Chapter 6 Statistical Inference 233
There are four elements in a hypothesis test:
- A null hypothesis, H 0
- An alternative hypothesis, Ha
- A test statistic
- A rejection region
The null hypothesis, usually labeled H 0 , represents the default or status
quo theory about the phenomenon that you’re studying. You accept the null
hypothesis as true unless you have convincing evidence to the contrary.
The alternative hypothesis, or Ha, represents an alternative theory that is
automatically accepted as true if the null hypothesis is rejected. Often the
alternative hypothesis is the hypothesis you want to accept. For example, a
new medication is being studied that claims to reduce blood pressure. The
null hypothesis is that the medication does not affect the patient’s blood
pressure. The alternative hypothesis is that the medication does affect the
patient’s blood pressure (in either a positive or a negative direction).
The test statistic is a statistic calculated from the data that you use to de-
cide whether to reject or to accept the null hypothesis. The rejection region
specifi es the set of values of the test statistic under which you’ll reject the
null hypothesis (and accept the alternative).
Types of Error
We can never be sure that our conclusions are free from error, but we can try
to reduce the probability of error. In hypothesis testing, we can make two
types of errors:
- Type I error: Rejecting the null hypothesis when the null hypothesis
is true
- Type II error: Failing to reject the null hypothesis when the alternative
hypothesis is true
The probability of Type I error is denoted by the Greek letter a, and the
probability of Type II error is identifi ed by the Greek letter b.
Generally, statisticians are more concerned with the probability of Type
I error, because rejecting the null hypothesis often results in some funda-
mental change in the status quo. In the blood pressure medication example,
incorrectly accepting the alternative hypothesis could result in prescribing an
ineffective drug to thousands of people. Statisticians will set a limit, called
the signifi cance level, that is the highest probability of Type I error allowed.
An accepted value for the signifi cance level is 0.05. This means we set up a
region that has probability .05 if the null hypothesis is true, and we reject H 0
if the data fall in this region.
Reducing Type II error becomes important in the design of experiments,
where the statistician wants to ensure that the study will detect an effect if a
true difference exists. An analysis of the probability of Type II error can aid
the statistician in determining how many subjects to have in the study.
