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Part IV: Guesstimating and Hypothesizing with Confidence
However, that doesn’t necessarily mean a real difference isn’t present in the
population of all students. But the researcher can’t say the computer game is
a better reading method based on this sample of 10 students. (See Chapter 14
for information on the power of a hypothesis test and its relationship to
sample size.)
In many paired experiments, the data sets are small due to costs and time
associated with doing these kinds of studies. That means the t-distribution
(see the t-table in the appendix) is often used instead of the standard normal
(Z-) distribution (the Z-table in the appendix) when figuring out the p-value.
Comparing Two Population Proportions
This test is used when the variable is categorical (for example, smoker/
nonsmoker, Democrat/Republican, support/oppose an opinion, and so
on) and you’re interested in the proportion of individuals with a certain
characteristic — for example, the proportion of smokers. In this case, two
populations or groups are being compared (such as the proportion of female
smokers versus male smokers).
In order to conduct this test, two independent (separate) random samples
need to be selected, one from each population. The null hypothesis is that the
two population proportions are the same; in other words, that their difference
is equal to 0. The notation for the null hypothesis is H : p = p , where p is the
o 1 2 1
proportion from the first population, and p is the proportion from the second
2
population.
Stating in H that the two proportions are equal is the same as saying their dif-
o
ference is zero. If you start with the equation p = p and subtract p from each
1 2 2
side, you get p – p = 0. So you can write the null hypothesis either way.
1 2
The formula for the test statistic comparing two proportions (under certain
conditions) is
where is the proportion in the first sample with the characteristic of interest,
is the proportion in the second sample with the characteristic of interest,
is the proportion in the combined sample (all the individuals in the first and
second samples together) with the characteristic of interest, and z is a value
on the Z-distribution (see Chapter 9). To calculate the test statistic, do the
following:

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