Mathematical Methods for Physics and Engineering : A Comprehensive Guide

31.7 HYPOTHESIS TESTING

t

P(t|H 0 )

tcrit

α

t

tcrit

P(t|H 1 )

β

Figure 31.10 The sampling distributionsP(t|H 0 )andP(t|H 1 ) of a test statistic

t. The shaded areas indicate the (one-tailed) regions for which Pr(t>tcrit|H 0 )=

αand Pr(t<tcrit|H 1 )=βrespectively.

random variable. Moreover, given the simple null hypothesisH 0 concerning the

PDF from which the sample was drawn, we may determine (in principle) the

sampling distributionP(t|H 0 ) of the test statistic. A typical example of such a

sampling distribution is shown in figure 31.10. One defines fortarejection region

containing some fractionαof the total probability. For example, the (one-tailed)

rejection region could consist of values oftgreater than some valuetcrit,for

which

Pr(t>tcrit|H 0 )=

∫∞

tcrit

P(t|H 0 )dt=α; (31.106)

this is indicated by the shaded region in the upper half of figure 31.10. Equally,

a (one-tailed) rejection region could consist of values oftless than some value

tcrit. Alternatively, one could define a (two-tailed) rejection region by two values

t 1 andt 2 such that Pr(t 1 <t<t 2 |H 0 )=α. In all cases, if the observed value oft

lies in the rejection region thenH 0 isrejectedatsignificance levelα;otherwiseH 0

isacceptedat this same level.

It is clear that there is a probabilityαof rejecting the null hypothesisH 0

even if it is true. This is called anerror of the first kind. Conversely, anerror

of the second kindoccurs when the hypothesisH 0 is accepted even though it is