Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

8.4.∗The Sequential Probability Ratio Test 505

can be rewritten, by taking logarithms, as

−log 9<

6

∑

xi− 459 n

200

<log 9.

This inequality is equivalent to the inequality

c 0 (n)=

153

2

n−

100

3

log 9<

∑n

1

xi<

153

2

n+

100

3

log 9 =c 1 (n).

Moreover,L(75,n)/L(78,n)≤k 0 andL(75,n)/L(78,n)≥k 1 are equivalent to the
inequalities

∑n

1 xi≥c^1 (n)and

∑n
1 xi≤c^0 (n), respectively. Thus the observation
of outcomes is discontinued with the first value ofnofNfor which either

∑n
1 xi≥
c 1 (n)or

∑n

1 xi ≤c^0 (n). The inequality

∑n
1 xi≥ c^1 (n) leads to the rejection
ofH 0 :θ= 75, and the inequality

∑n
1 xi ≤c^0 (n) leads to the acceptance of
H 0 :θ= 75. The power of the test is approximately 0.10 whenH 0 is true, and
approximately 0.90 whenH 1 is true.

Remark 8.4.2.It is interesting to note that a sequential probability ratio test can

be thought of as arandom-walk procedure. To illustrate, the final inequalities of

Examples 8.4.1 and 8.4.2 can be written as

−log 2 k 1 <

∑n

1

2(xi− 0 .5)<−log 2 k 0

and

−

100

3

log 9<

∑n

1

(xi− 76 .5)<

100

3

log 9,

respectively. In each instance, think of starting at the point zero and taking random

steps until one of the boundaries is reached. In the first situation the random steps

are 2(X 1 − 0 .5),2(X 2 − 0 .5),2(X 3 − 0 .5),..., which have the same length, 1, but

with random directions. In the second instance, both the length and the direction

of the steps are random variables,X 1 − 76. 5 ,X 2 − 76. 5 ,X 3 − 76. 5 ,....

In recent years, there has been much attention devoted to improving quality
of products using statistical methods. One such simple method was developed by
Walter Shewhart in which a sample of sizenof the items being produced is taken and
they are measured, resulting innvalues. The meanXof thesenmeasurements has
an approximate normal distribution with meanμand varianceσ^2 /n. In practice,μ
andσ^2 must be estimated, but in this discussion, we assume that they are known.
From theory we know that the probability is 0.997 thatxis between

LCL =μ−

3 σ

√

n

and UCL =μ+

3 σ

√

n

.

These two values are called the lower (LCL) and upper (UCL) control limits, respec-

tively. Samples like these are taken periodically, resulting in a sequence of means,