4.5. Introduction to Hypothesis Testing 273Table 4.5.2:Plant GrowthPot 12345678
Cross 23.500 12.000 21.000 22.000 19.125 21.500 22.125 20.375
Self 17.375 20.375 20.000 20.000 18.375 18.625 18.625 15.250
Pot 9 101112131415
Cross 18.250 21.625 23.250 21.000 22.125 23.000 12.000
Self 16.500 18.000 16.250 18.000 12.750 15.500 18.000outliers, normality of the error distribution is somewhat dubious, and we use the
test in a conservative manner, as discussed at the end of Example 4.5.4.
Assuming that the rda filedarwin.rdahas been loaded in R, the code for the
abovet-test ist.test(cross-self,mu=0,alt="greater")which evaluates thet-
test statistic to be 2.1506.Difference= Cross – Self- 5
0510 Panel ADifference= Cross – Self- 5
0510 Panel B- 1.5–1.0–0.5 0.0
Normal quantiles0.5 1.0 1.5
Figure 4.5.2:Boxplot and normalq−qplot for the data of Example 4.5.5.EXERCISES
In many of these exercises, use R or another statistical package for computations
and graphs of power functions.
4.5.1. Show that the approximate power function given in expression (4.5.12) of
Example 4.5.3 is a strictly increasing function ofμ. Show then that the test discussed
in this example has approximate sizeαfor testing
H 0 : μ≤μ 0 versusH 1 : μ>μ 0.4.5.2.For the Darwin data tabled in Example 4.5.5, verify that the Studentt-test
statistic is 2.15.4.5.3.LetXhave a pdf of the formf(x;θ)=θxθ−^1 , 0 <x<1, zero elsewhere,
whereθ∈{θ:θ=1, 2 }. To test the simple hypothesisH 0 :θ= 1 against the
alternative simple hypothesisH 1 :θ= 2, use a random sampleX 1 ,X 2 of sizen=2