4.6. Additional Comments About Statistical Tests 281For thep-value, compute each of the one-sidedp-values, take the smallerp-value, and
double it. For an illustration, in the Darwin example, suppose the the hypotheses
areH 0 :μ=0versusH 1 :μ = 0. Then thep-value is 2(0.0248) = 0.0496. As
a final note onp-values for two-sided hypotheses, suppose the test statistic can
be expressed in terms of at-test statistic. In this case thep-value can be found
equivalently as follows. Ifdis the realized value of thet-test statistic then the
p-value is
p-value =PH 0 [|t|≥|d|], (4.6.15)
where, underH 0 ,thas at-distribution withn−1 degrees of freedom.
In this discussion onp-values, keep in mind that good science dictates that the
hypotheses should be known before the data are drawn.EXERCISES4.6.1.The R functionzpower, found at the site listed in the Preface, computes
the plot in Figure 4.6.1. Consider the two-sided test for proportions discussed in
Example 4.6.3 based on the test statisticZ 1. Specifically consider the hypotheses
H 0 :p=. 0 .6versusH 1 :p =0.6. Using the sample sizen= 50 and the level
α=0.05, write a R program, similar tozpower, which computes a plot of the
power curve for this test on a proportion.4.6.2.Consider the power functionγ(μ) and its derivativeγ′(μ) given by (4.6.5)
and (4.6.6). Show thatγ′(μ) is strictly negative forμ<μ 0 and strictly positive for
μ>μ 0.
4.6.3.Show that the test defined by 4.6.9 has exact sizeαfor testingH 0 : μ=μ 0
versusH 1 :μ =μ 0.4.6.4.Consider the one-sidedt-test forH 0 :μ=μ 0 versusHA 1 :μ>μ 0 con-
structed in Example 4.5.4 and the two-sidedt-test fort-test forH 0 :μ=μ 0 versus
H 1 : μ =μ 0 given in (4.6.9). Assume that both tests are of sizeα. Show that for
μ>μ 0 , the power function of the one-sided test is larger than the power function
of the two-sided test.
4.6.5.On page 373 Rasmussen (1992) discussed a paired design. A baseball coach
paired 20 members of his team by their speed; i.e., each member of the pair has
about the same speed. Then for each pair, he randomly chose one member of the
pair and told him that if could beat his best time in circling the bases he would
give him an award (call this response the time of the “self” member). For the other
member of the pair the coach’s instruction was an award if he could beat the time
of the other member of the pair (call this response the time of the “rival” member).
Each member of the pair knew who his rival was. The data are given below, but are
also in the fileselfrival.rda.Letμdbe the true difference in times (rival minus
self) for a pair. The hypotheses of interest areH 0 :μd=0versusH 1 :μd<0. The
data are in order by pairs, so do not mix the order.
self: 16.20 16.78 17.38 17.59 17.37 17.49 18.18 18.16 18.36 18.53