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276 Chapter 8/Statistical intervals for a single sample

E = error = x –

FIGURE 8-2 Error in l = x – z /2 / n x u = x + z /2 / n
estimating μ with x.
related to the coni dence level. It is desirable to obtain a coni dence interval that is short
enough for decision-making purposes and that also has adequate coni dence. One way to
achieve this is by choosing the sample size n to be large enough to give a CI of specii ed length
or precision with prescribed coni dence.

8-1.2 CHOICE OF SAMPLE SIZE

The precision of the coni dence interval in Equation 8-5 is 2z α σ n. This means that in
2 /
using x to estimate μ, the error E = x − μ is less than or equal to z α σ n with coni dence
/2
100 1 − ≠ ). This is shown graphically in Fig. 8-2. In situations whose sample size can be con-
(
trolled, we can choose n so that we are 100 1( − ≠ )% conident that the error in estimating μ is

less than a speciied bound on the error E. The appropriate sample size is found by choosing n

such that z α σ n = E. Solving this equation gives the following formula for n.
/2
(
Sample Size for If x is used as an estimate of μ, we can be 100 1 − ≠ )% coni dent that the error

Speciied Error on the | x − μ | will not exceed a specii ed amount E when the sample size is
Mean, Variance Known ⎛ z / σ⎞ 2
n = ⎜ ⎝ α 2 ⎟ (8-6)
E ⎠

If the right-hand side of Equation 8-6 is not an integer, it must be rounded up. This will ensure
that the level of conidence does not fall below 100 1 − ≠ )%. Notice that 2E is the length of
(

the resulting coni dence interval.

Example 8-2 Metallic Material Transition To illustrate the use of this procedure, consider the CVN test
described in Example 8-1 and suppose that we want to determine how many specimens must be
tested to ensure that the 95% CI on μ for A238 steel cut at 60°C has a length of at most 1 0. J. Because the bound on
error in estimation E is one-half of the length of the CI, to determine n, we use Equation 8-6 with E = 0 5, σ = 1, and
.
z α = . . The required sample size is,
1 96
/2
) ⎤
⎛ σ⎞ 2 . ( ⎡ 1 96 1 2
n = ⎜ ⎝ z α/2 ⎟ = ⎢ ⎣ . 0 5 ⎥ = 15 .37
E ⎠
⎦
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 coni dence inter-
val 2E, coni dence level 100 1( − ≠ ), and standard deviation σ:

r As the desired length of the interval 2E decreases, the required sample size n increases for
a ixed value of σ and specii ed coni dence.

r As σ increases, the required sample size n increases for a ixed desired length 2E and speci-
i ed coni dence.

r As the level of conidence increases, the required sample size n increases for i xed desired
length 2E and standard deviation σ.