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Unit 31: Hypothesis Testing

The two types of errors are shown by the following figure. Notes

Figure 31.1

It is obvious, from the above figure, that it is not possible to simultaneously control both types
of errors because a decrease in probability of committing one type of error is accompanied by
the increase in probability of committing the other type of error. Further, we may note that
farther the true value of parameter from the hypothesised value, smaller would be the size of
type II error, b. The graph of various values of m against b is known as the Operating Characteristic
Surve.

In the procedure of testing a hypothesis, the probability or size of type I error, i.e., a is specified
in advance. Usually we take a = 0.05 ( i.e., 5%) or 0.01 (i.e., 1%). Also see remarks (1) given at the
end of this section.

Power of a Test

The power of a test is defined as the probability of rejecting a false null hypothesis. Since b is the
probability of accepting a false hypothesis, the power of test is given by 1 - b. More precisely, we
can write
Power of a test = P [Rejecting H /H is false] = 1 – b
0 0
Since the value of  depends upon the true value of population parameter ( in the above
example), the relationship between various values of m and 1 -  is termed as power function, as
shown in Figure 31.2.

Figure 31.2

Critical Region and One Tailed versus Two Tailed Tests

Let H :  =  against H :    , where  denotes some specified value of population mean m.
0 0 a 0 0
For example,  = 1600, in the example considered above.
0

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