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304 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

Finally, most computer programs report P-values along with the computed value of the
test statistic. Some hand-held calculators also have this capability. In Example 9-6, Minitab
gave the P-value for the value t 0 2.72 in Example 9-6 as 0.008.
9-3.3 Choice of Sample Size

The type II error probability for tests on the mean of a normal distribution with unknown vari-
ance depends on the distribution of the test statistic in Equation 9-23 when the null hypothe-
sis H 0 : 0 is false. When the true value of the mean is 0 , the distribution for T 0
is called the noncentral t distribution with n
1 degrees of freedom and noncentrality pa-
rameter 1n
. Note that if 0, the noncentral t distribution reduces to the usual central
t distribution. Therefore, the type II error of the two-sided alternative (for example) would be

P5
t
2,n
1 T t
2,n
1 0 06
0
P5
t
2,n
1 T¿ t
2,n
1 6
0
where T¿ 0 denotes the noncentral t random variable. Finding the type II error probability for
the t-test involves finding the probability contained between two points of the noncentral t
distribution. Because the noncentral t-random variable has a messy density function, this in-
tegration must be done numerically.
Fortunately, this ugly task has already been done, and the results are summarized in a se-
ries of O.C. curves in Appendix Charts VIe, VIf, VIg, and VIh that plot for the t-test against
a parameter d for various sample sizes n. Curves are provided for two-sided alternatives on
Charts VIe and VIf. The abscissa scale factor d on these charts is defined as
0
0 0 0
0
d (9-24)

For the one-sided alternative 0 or 0 , we use charts VIg and VIh with

0
0 0 0
0
d (9-25)

2
We note that d depends on the unknown parameter . We can avoid this difficulty in sev-
eral ways. In some cases, we may use the results of a previous experiment or prior information
2
to make a rough initial estimate of . If we are interested in evaluating test performance after
2
2
the data have been collected, we could use the sample variance s to estimate . If there is no
2
previous experience on which to draw in estimating , we then define the difference in the
mean d that we wish to detect relative to . For example, if we wish to detect a small difference
in the mean, we might use a value of d 0 0
1 (for example), whereas if we are interested
in detecting only moderately large differences in the mean, we might select d 0 0
2 (for
example). That is, it is the value of the ratio 0 0
that is important in determining sample size,
and if it is possible to specify the relative size of the difference in means that we are interested in
detecting, then a proper value of d can usually be selected.
EXAMPLE 9-7 Consider the golf club testing problem from Example 9-6. If the mean coefficient of restitu-
tion exceeds 0.82 by as much as 0.02, is the sample size n 15 adequate to ensure that H :
0
0.82 will be rejected with probability at least 0.8?
To solve this problem, we will use the sample standard deviation s 0.02456 to estimate
. Then d 0 0
0.02
0.02456 0.81 . By referring to the operating characteristic
curves in Appendix Chart VIg (for 0.05) with d 0.81 and n 15, we find that 0.10,