272 Some Elementary Statistical InferencesSo if we have some reasonable idea of whatσequals, we can compute the approxi-
mate power function. As Exercise 4.5.1 shows, this approximate power function is
strictly increasing inμ, so as in the last example, we can change the null hypotheses
to
H 0 : μ≤μ 0 versusH 1 : μ>μ 0. (4.5.13)Our asymptotic test has approximate sizeαfor these hypotheses.
Example 4.5.4(Test forμUnder Normality).LetXhave aN(μ, σ^2 ) distribution.
As in Example 4.5.3, consider the hypothesesH 0 : μ=μ 0 versusH 1 : μ>μ 0 , (4.5.14)whereμ 0 is specified. Assume that the desired size of the test isα,for0<α<1,
SupposeX 1 ,...,Xnis a random sample from aN(μ, σ^2 ) distribution. LetXand
S^2 denote the sample mean and variance, respectively. Our intuitive rejection rule
is to rejectH 0 in favor ofH 1 ifXis much larger thanμ 0. Unlike Example 4.5.3, we
now know the distribution of the statisticX. In particular, by Part (d) of Theorem
3.6.1, underH 0 the statisticT=(X−μ 0 )/(S/
√
n)hasat-distribution withn− 1
degrees of freedom. Using the distribution ofT, it follows that this rejection rule
has exact levelα:RejectH 0 in favor ofH 1 ifT=XS/−√μn^0 ≥tα,n− 1 , (4.5.15)wheretα,n− 1 is the upperαcritical point of at-distribution withn−1 degrees of
freedom; i.e.,α=P(T>tα,n− 1 ). This is often called thet-testofH 0 :μ=μ 0.
Note the differences between this rejection rule and the large sample rule, (4.5.11).
The large sample rule has approximate levelα, while this has exact levelα.Of
course, we now have to assume thatXhas a normal distribution. In practice, we
may not be willing to assume that the population is normal. Usuallyt-critical val-
ues are larger thanz-critical values; hence, thet-test is conservative relative to the
large sample test. So, in practice, many statisticians often use thet-test.
The R codet.test(x,mu=mu0,alt="greater")computes thet-test for the hy-
potheses (4.5.14), where the R vectorxcontains the sample.
Example 4.5.5(Example 4.5.1, Continued).The data for Darwin’s experiment
onZea maysare recorded in Table 4.5.2 and are, also, in the filedarwin.rda.A
boxplot and a normalq−qplot of the 15 differences,xi=yi−zi, are found in
Figure 4.5.2. Based on these plots, we can see that there seem to be two outliers,
Pots 2 and 15. In these two pots, the self-fertilizedZea maysare much taller than
their cross-fertilized pairs. Except for these two outliers, the differences,yi−zi,are
positive, indicating that the cross-fertilization leads to taller plants. We proceed
to conduct a test of hypotheses (4.5.2), as discussed in Example 4.5.4. We use the
decision rule given by (4.5.15) withα=0.05. As Exercise 4.5.2 shows, the values
of the sample mean and standard deviation for the differences,xi,arex=2. 62
andsx=4.72. Hence thet-test statistic is 2.15, which exceeds thet-critical value,
t. 05 , 14 =qt(0.95,14)=1.76. Thus we rejectH 0 and conclude that cross-fertilized
Zea maysare on the average taller than self-fertilizedZea mays. Because of the