sample is often large enough so as to produce negligible sampling error inp 0 ).
Or we could be concerned with a research question such as: Does the side e¤ect
(of a certain drug) exceed regulated limitp 0? In Exercise 5.5 we compared
the incidence of breast cancer among female beauticians (who are frequently
exposed to the use of hair dyes) versus a standard level of 7/1000 (for five years)
for ‘‘average’’ American women. The figure 7/1000 isp 0 for that example.
In a typical situation, the null hypothesis of a statistical test is concerned
with a parameter; the parameter in this case is the proportionp. Sample data
are summarized into a statistic used to estimate the parameter under investiga-
tion. Since the parameter under investigation is the proportionp, our focus in
this case is the sample proportionp. In general, a statistic is itself a variable
with a specific sampling distribution (in the context of repeated sampling). Our
statistic in this case is the sample proportion p; the corresponding sampling
distribution is obtained easily by invoking thecentral limit theorem. With large
sample size and assuming that the null hypothesisH 0 is true, it is the normal
distribution with mean and variance given by
mp¼p 0sp^2 ¼
p 0 ð 1 p 0 Þ
nrespectively. From this sampling distribution, the observed value of the sample
proportion can be converted to a standard unit: the number of standard errors
away from the hypothesized value ofp 0. In other words, to perform a test of
significance forH 0 , we proceed with the following steps:
- Decide whether a one- or a two-sided test is appropriate.
- Choose a level of significancea, a common choice being 0.05.
- Calculate thezscore
z¼pp 0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p 0 ð 1 p 0 Þ=np- From the table for the standard normal distribution (Appendix B) and
the choice ofa(e.g.,a¼ 0 :05), the rejection region is determined by:
(a) For a one-sided test:
za 1 : 65 forHA:p<p 0
zb 1 : 65 forHA:p>p 0(b) For a two-sided test orHA:p 0 p 0 :za 1 :96 or zb 1 : 96ONE-SAMPLE PROBLEM WITH BINARY DATA 209