Introduction to Probability and Statistics for Engineers and Scientists

298 Chapter 8:Hypothesis Testing

1.0

0.8

0.6

0.4

0.2

0

Probability of accepting

H^0

n

12345 d = |σm − m^0 |

.95

FIGURE 8.2 The OC curve for the two-sided normal test for significance levelα=.05.

Asz.025=1.96, the desired probability is, from Equation 8.3.4,

(−

√

5 +1.96)− (−

√

5 −1.96)

= 1 − (

√

5 −1.96)−[ 1 − (

√

5 +1.96)]

= (4.196)− (.276)

=.392 ■

REMARK

The function 1−β(μ) is called thepower-functionof the test. Thus, for a given valueμ,
the power of the test is equal to the probability of rejection whenμis the true value. ■

The operating characteristic function is useful in determining how large the random
sample need be to meet certain specifications concerning type II errors. For instance,
supposethatwedesiretodeterminethesamplesizennecessarytoensurethattheprobability
of acceptingH 0 :μ=μ 0 when the true mean is actuallyμ 1 is approximatelyβ. That is,
we wantnto be such that

β(μ 1 )≈β

But from Equation 8.3.4, this is equivalent to

(√

n(μ 0 −μ 1 )

σ

+zα/2

)

−

(√

n(μ 0 −μ 1 )

σ

−zα/2

)

≈β (8.3.5)

Although the foregoing cannot be analytically solved forn, a solution can be obtained by
using the standard normal distribution table. In addition, an approximation forncan be
derived from Equation 8.3.5 as follows. To start, suppose thatμ 1 >μ 0. Then, because
this implies that

μ 0 −μ 1

σ/

√

n

−zα/2≤−zα/2