Applied Statistics and Probability for Engineers

9-1 HYPOTHESIS TESTING 281

48.5 50 51.5

Reject H 0

μ ≠ 50 cm/s

Fail to Reject H 0

μ = 50 cm/s

Reject H 0

μ ≠ 50 cm/s

x

Table 9-1 Decisions in Hypothesis Testing

Decision H 0 Is True H 0 Is False

Fail to reject H 0 no error type II error

Figure 9-1 Decision criteria for testing H 0 :  Reject H 0 type I error no error

50 centimeters per second versus H 1 : 50 centime-

ters per second.

α/2 = 0.0287 α/2 = 0.0287

48.5 μ= 50 51.5 X

Figure 9-2 The critical region for H 0 :  50

versus H 1 : 50 and n10.

Because our decision is based on random variables, probabilities can be associated with

the type I and type II errors in Table 9-1. The probability of making a type I error is denoted

by the Greek letter. That is,

    P(type I error)P(reject H 0 when H 0 is true) (9-3)

Sometimes the type I error probability is called the significance level,or the-error,or the

size of the test. In the propellant burning rate example, a type I error will occur when either

or when the true mean burning rate is centimeters per second.

Suppose that the standard deviation of burning rate is centimeters per second and that

the burning rate has a distribution for which the conditions of the central limit theorem apply,

so the distribution of the sample mean is approximately normal with mean and stan-

dard deviation. The probability of making a type I error (or the

significance level of our test) is equal to the sum of the areas that have been shaded in the tails

of the normal distribution in Fig. 9-2. We may find this probability as

The z-values that correspond to the critical values 48.5 and 51.5 are

Therefore

This implies that 5.76% of all random samples would lead to rejection of the hypothesis

when the true mean burning rate is really 50 centimeters

per second.

H 0 : 50 centimeters per second

    P 1 Z

1.90 2 P 1 Z1.90 2 0.028717 0.0287170.057434

z 1 

48.5 

50

0.79



1.90 and z 2 

51.5 

50

0.79

1.90

    P 1 X48.5 when  502 P 1 X51.5 when  502


1 n2.5
110 0.79

 50

2.5

x51.5 x48.5  50

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