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7.5 HYPOTHESIS TESTING: A SINGLE POPULATION PROPORTION 259
sample sufficiently large for application of the central limit theorem as discussed in Section
5.5 is available for analysis, the test statistic is
z = pN - p 0
p 0 q 0
A n
which, when H 0 is true, is distributed approximately as the standard normal.
(7.5.1)
EXAMPLE 7.5.1
Wagenknecht et al. (A-20) collected data on a sample of 301 Hispanic women living in
San Antonio, Texas. One variable of interest was the percentage of subjects with impaired
fasting glucose (IFG). IFG refers to a metabolic stage intermediate between normal glucose
homeostasis and diabetes. In the study, 24 women were classified in the IFG stage.
The article cites population estimates for IFG among Hispanic women in Texas as 6.3
percent. Is there sufficient evidence to indicate that the population of Hispanic women
in San Antonio has a prevalence of IFG higher than 6.3 percent
Solution:
1. Data. The data are obtained from the responses of 301 individuals
of which 24 possessed the characteristic of interest; that is, Np =
24>301 = .080.
2. Assumptions. The study subjects may be treated as a simple random
sample from a population of similar subjects, and the sampling distribution
of Np is approximately normally distributed in accordance with
the central limit theorem.
3. Hypotheses.
H 0 : p … .063
H A : p 7 .063
We conduct the test at the point of equality. The conclusion we reach
will be the same as we would reach if we conducted the test using any
other hypothesized value of p greater than .063. If H 0 is true, p = .063
and the standard error s pN = 11.06321.9372>301. Note that we use the
hypothesized value of p in computing s pN . We do this because the
entire test is based on the assumption that the null hypothesis is true.
To use the sample proportion, pN , in computing s pN would not be consistent
with this concept.
4. Test statistic. The test statistic is given by Equation 7.5.1.
5. Distribution of test statistic. If the null hypothesis is true, the test statistic
is approximately normally distributed with a mean of zero.
6. Decision rule. Let a = .05. The critical value of z is 1.645. Reject
if the computed z is Ú1.645.
H 0