Page 300 - Applied Statistics And Probability For Engineers

c08.qxd 5/16/02 11:35 M Page 252 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files:

252 CHAPTER 8 STATISTICAL INTERVALS FOR A SINGLE SAMPLE

µ
E = error = x – 
Figure 8-2 Error in
estimating with . x l = x – z α σ / n x µ u = x + z α σ / n
/2
/2
The length of a confidence interval is a measure of the precision of estimation. From the
preceeding discussion, we see that precision is inversely related to the confidence level. It is de-
sirable to obtain a confidence interval that is short enough for decision-making purposes and
that also has adequate confidence. One way to achieve this is by choosing the sample size n to
be large enough to give a CI of specified length or precision with prescribed confidence.

8-2.2 Choice of Sample Size

The precision of the confidence interval in Equation 8-7 is 2z 2 1n. This means that in
using x to estimate , the error E 0 x 0 is less than or equal to z 2 1n with
confidence 100(1 ). This is shown graphically in Fig. 8-2. In situations where the sam-
ple size can be controlled, we can choose n so that we are 100(1 ) percent confident that
the error in estimating is less than a specified bound on the error E. The appropriate sam-
1n E. Solving this equation gives the fol-
ple size is found by choosing n such that z 2
lowing formula for n.

Definition
If is used as an estimate of , we can be 100(1 )% confident that the error
x
0 x 0 will not exceed a specified amount E when the sample size is

2
z 2
n a b (8-8)
E

If the right-hand side of Equation 8-8 is not an integer, it must be rounded up. This will ensure
that the level of confidence does not fall below 100(1 )%. Notice that 2E is the length of
the resulting confidence interval.
EXAMPLE 8-2 To illustrate the use of this procedure, consider the CVN test described in Example 8-1, and
suppose that we wanted to determine how many specimens must be tested to ensure that the
95% CI on for A238 steel cut at 60°C has a length of at most 1.0J. Since the bound on error
in estimation E is one-half of the length of the CI, to determine n we use Equation 8-8 with
E 0.5, 1, and z 2 0.025. The required sample size is 16

z 2 2 11.9621 2
n a b c d 15.37
E 0.5

and because n must be an integer, the required sample size is n 16.

Notice the general relationship between sample size, desired length of the confidence
interval 2E, confidence level 100(1 ), and standard deviation :

As the desired length of the interval 2E decreases, the required sample size n increases
for a fixed value of and specified confidence.