Page 424 - DMTH404_STATISTICS

Statistics

Notes
If we decide to have a = 0.05, we know that for a standard normal variate P[- 1.96 £ z £ 1.96] =
1 - 0.05 = 0.95, the procedure of testing of hypothesis can be outlined as:

æ X  ö
-
Reject H if the computed value of z from the sample . ., i e z = lies outside the interval
0 ç cal ÷
è s / n ø
(- 1.96, 1.96) and accept it otherwise.
In terms of figure, the portion of z axis covering the interval (- 1.96, 1.96), i.e, A to B is termed as
the Acceptance Region and its remaining portions, which lie to the left of point A and to the right
of point B, are termed as the Region of Rejection or Critical Region (C.R.).

Figure 31.3

The specification of the critical region for a test depends upon the nature of the alternative
hypothesis and the value of . For example, H :    , this implies that m may be less or greater
a 0
than  . Thus, the critical region is to be specified on both tails of the curve with each part
0
corresponding to half of the value of a. A test having critical region at both the tails of the
probability curve is termed as a two tailed test.
Further, if H :  >  or  <  , the critical region is to be specified only at one tail of the
a 0 0
probability curve and the corresponding test is termed as a one tailed test. These situations are
shown in the following figures.
The values of the random variable separating the acceptance region from critical region are
termed as critical value(s). For example, z and z , shown above, are critical values. Similarly,
a/2 a
for a normal distribution the critical values for a two tailed test are - 1.96 and 1.96 for  = 0.05 or
- 2.58 and 2.58 for  = 0.01 and the corresponding value for a one tailed test is ± 1.645 or ± 2.33
depending upon whether a = 0.05 or 0.01.

Figure 31.4

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