4.6. Additional Comments About Statistical Tests 2754.5.11.LetY 1 <Y 2 <Y 3 <Y 4 be the order statistics of a random sample of size
n= 4 from a distribution with pdff(x;θ)=1/θ, 0 <x<θ, zero elsewhere, where
0 <θ.ThehypothesisH 0 :θ= 1 is rejected andH 1 :θ>1 is accepted if the
observedY 4 ≥c.
(a)Find the constantcso that the significance level isα=0.05.(b)Determine the power function of the test.4.5.12. LetX 1 ,X 2 ,...,X 8 be a random sample of sizen= 8 from a Poisson
distribution with meanμ. Reject the simple null hypothesisH 0 :μ=0.5and
acceptH 1 :μ> 0 .5 if the observed sum
∑ 8
i=1xi≥8.
(a)Show that the significance level is1-ppois(7,8*.5).(b)Use R to determineγ(0.75),γ(1), andγ(1.25).(c)Modify the code in Exercise 4.5.9 to obtain a plot of the power function.4.5.13.Letpdenote the probability that, for a particular tennis player, the first
serve is good. Sincep=0.40, this player decided to take lessons in order to increase
p. When the lessons are completed, the hypothesisH 0 :p=0.40 is tested against
H 1 :p> 0 .40 based onn=25trials. LetY equal the number of first serves that
are good, and let the critical region be defined byC={Y:Y≥ 13 }.
(a)Show thatαis computed byα=1-pbinom(12,25,.4).(b)Findβ=P(Y<13) whenp=0.60; that is,β=P(Y ≤12;p=0.60) so
that 1−βis the power atp=0.60.4.5.14.LetSdenote the number of success inn= 40 Bernoulli trials with prob-
ability of successp. Consider the hypotheses:H 0 :p≤ 0 .3versusH 1 :p> 0 .3.
Consider the two tests: (1) RejectH 0 ifS≥16 and (2) RejectH 0 ifS≥17.
Determine the level of these tests. The R functionbinpower.rproduces a version
of Figure 4.5.1. For this exercise, write a similar R function that graphs the power
functions of the above two tests.
4.6 Additional Comments About Statistical Tests
All of the alternative hypotheses considered in Section 4.5 wereone-sided hy-
potheses. For illustration, in Exercise 4.5.8 we testedH 0 :μ=30,000 against the
one-sided alternativeH 1 :μ> 30 ,000, whereμis the mean of a normal distribution
having standard deviationσ= 5000. Perhaps in this situation, though, we think
the manufacturer’s process has changed but are unsure of the direction. That is,
we are interested in the alternativeH 1 :μ =30,000. In this section, we further ex-
plore hypotheses testing and we begin with the construction of a test for a two-sided
alternative.