Applied Statistics and Probability for Engineers

normal.The important result upon which the test procedure relies is that if X 1 , X 2 , p,Xnis a

random sample from a normal distribution with mean and variance 2 , the random variable

has a tdistribution with n 1 degrees of freedom. Recall that we used this result in Section

8-3 to devise the t-confidence interval for . Now consider testing the hypotheses

We will use the test statistic

(9-23)

If the null hypothesis is true, T 0 has a tdistribution with n 1 degrees of freedom. When we

know the distribution of the test statistic when H 0 is true (this is often called the reference

distributionor the null distribution), we can locate the critical region to control the type I

error probability at the desired level. In this case we would use the tpercentage points t 2,n 

1

and as the boundaries of the critical region so that we would reject H 0 :   0 if

where t 0 is the observed value of the test statistic T 0. The test procedure is very similar to the

test on the mean with known variance described in Section 9-2, except that T 0 is used as the

test statistic instead of Z 0 and the tn 

1 distribution is used to define the critical region instead

of the standard normal distribution. A summary of the test procedures for both two- and one-

sided alternative hypotheses follows:

t 0 t 2,n

1 or if t 0 
t 2,n
1

t 2,n 

1

T 0 

X

 0

S 

1 n

H 1 :  0

H 0 :  0

T

X



S 

1 n

9-3 TESTS ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE UNKNOWN 301

Figure 9-8 The reference distribution for H 0 :   0 with critical region for (a) (b) and

(c) H 1 :  0.

H 1 : Z 0 , H 1 :  0 ,

(a)

0

tn – 1

• t /α 2, n – 1 t /α 2, n – 1 tα , n – 1 – t α , n – 1 T 0

/2α /2α

(c)

0

α

(b)

0

α

tn – 1 tn – 1

Null hypothesis: H 0 :  0

Test statistic:

Alternative hypothesis Rejection criteria

H 1 :  0 t 0 

t ,n 

1

H 1 :  0 t 0 t ,n 

1

H 1 : Z 0 t 0 t /2,n
1 or t 0 
t /2,n
1

T 0 

X

 0

S 

1 n

The One-

Sample t-Test

Figure 9-8 shows the location of the critical region for these situations.

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