Applied Statistics and Probability for Engineers

P-value approach is used, step 6 of the hypothesis-testing procedure can be modified.

Specifically, it is not necessary to state explicitly the critical region.

9-2.3 Connection between Hypothesis Tests and Confidence Intervals

There is a close relationship between the test of a hypothesis about any parameter, say , and

the confidence interval for . If [l,u] is a % confidence interval for the parameter

, the test of size of the hypothesis

will lead to rejection of H 0 if and only if  0 is notin the % CI [l,u]. As an illus-

tration, consider the escape system propellant problem discussed above. The null hypothesis

H 0 :  50 was rejected, using. The 95% two-sided CI on can be calculated using

Equation 8-7. This CI is 50.52   52.08. Because the value  0  50 is not included in this

interval, the null hypothesis H 0 :  50 is rejected.

Although hypothesis tests and CIs are equivalent procedures insofar as decision mak-

ing or inferenceabout is concerned, each provides somewhat different insights. For

instance, the confidence interval provides a range of likely values for at a stated confi-

dence level, whereas hypothesis testing is an easy framework for displaying the risk levels

such as the P-value associated with a specific decision. We will continue to illustrate the

connection between the two procedures throughout the text.

9-2.4 Type II Error and Choice of Sample Size

In testing hypotheses, the analyst directly selects the type I error probability. However, the

probability of type II error depends on the choice of sample size. In this section, we will

show how to calculate the probability of type II error . We will also show how to select the

sample size to obtain a specified value of .

Finding the Probability of Type II Error 

Consider the two-sided hypothesis

Suppose that the null hypothesis is false and that the true value of the mean is ,

say, where. The test statistic Z 0 is

Therefore, the distribution of Z 0 when H 1 is true is

Z 0 N a (9-16)

 1 n

, 1b

Z 0 

X

 0



1 n



X 

1  0  2



1 n

 1 n

 0

 0 

H 1 :  0

H 0 :  0

     0.05

100 11

    2

H 1 :  0

H 0 :  0

10011

    2

9-2 TESTS ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWN 293

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