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Mathematics Semester 3 STPM Chapter 5 Hypothesis Testing
Test statistics

A test statistic is a random variable whose value is used to determine whether a null hypothesis is rejected
in a hypothesis test.
The choice of a test statistic depends on the assumed probability distribution and the hypothesis under
question.
Consider the following example. A study claims that 20% drivers in a city involve running red lights. We
choose, at random, 20 drivers from the city. If more than 7 drivers admit to running red lights, it indicates
a higher percentage. In this case, we are essentially testing the null hypothesis that 20% drivers involve
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running red lights against the alternative hypothesis that the percentage is higher.
This can be written as follows:
H : p = 0.2,
0
H : p . 0.2.
1
The test statistic on which we base our decision is random variable X, the number of drivers in our test. The
possible values of X, from 0 to 20, are divided into two groups: those numbers less than or equal to 7 and
those greater than 7. All possible values obtained greater than 7 constitute what is called the critical region.
The set of values that leads to the rejection of H in favour of H is called a critical region or rejection region.
0
1
Thus, if x . 7, we reject H in favour of the alternative hypothesis H . If x < 7, we fail to reject H .
0
0
1
This decision criterion is illustrated in the figure below.
Critical region
Do not reject H 0 Reject H 0
(p = 0.2) (p > 0.2)
x
0 20
Critical value, 7

Figure 5.1

Type I and Type II errors

The decision procedure described in test statistics above could lead to either of two wrong conclusions. We
may reject H when in fact H is true, that is, the percentage of running of red lights by drivers does not
0
0
increase. This may occur because we happen to choose this particular selected group of drivers who have
such rude behaviour. Or, alternatively, we may not reject H when in fact H is false, that is, running red
0
0
lights is getting worse. 5
A Type I error occurs when a true H is rejected; a Type II error occurs when a false H is not rejected.
0 0
It is obvious that we cannot completely avoid making these errors. Our goal is try to keep the probability
of making these errors relatively small.
In testing any statistical hypothesis, there are four possible outcomes that determine whether our decision
is correct or in error. These four possibilities are listed in the following table.
H is true H is false
0 0
Do not reject H 0 Correct decision Type II error
Reject H Type I error Correct decision
0
Table 5.1

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