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The formula for the test statistic for a single proportion (under certain
conditions) is:
where is the proportion of individuals in the sample who have that charac-
teristic and z is a value on the Z-distribution (see Chapter 9). To calculate the
test statistic, do the following:
1. Calculate the sample proportion, , by taking the number of people in
the sample who have the characteristic of interest (for example, the
number of people in the sample carrying cellphones) and dividing
that by n, the sample size.
2. Find
, where p is the value in H .
o
o
3. Calculate the standard error,
Chapter 15: Commonly Used Hypothesis Tests: Formulas and Examples 233
.
4. Divide your result from Step 2 by your result from Step 3.
To interpret the test statistic, look up your test statistic on the standard
normal (Z-) distribution (in the appendix) and calculate the p-value (see
Chapter 14 for more on p-value calculations).
The conditions for using this test statistic are that
(see Chapter 9 for details).
For example, suppose Cavifree claims that four out of five dentists recom-
mend Cavifree toothpaste to their patients. In this case, the population is all
dentists, and p is the proportion of all dentists who recommended Cavifree.
The claim is that p is equal to “four out of five,” or p is 4 ÷ 5 = 0.80. You sus-
o
pect that the proportion is actually less than 0.80. Your hypotheses are H :
o
p = 0.80 versus H : p < 0.80.
a
Suppose that 151 out of your sample of 200 dental patients reported receiving
a recommendation for Cavifree from their dentist. To find the test statistic for
these results, follow these steps:
1. Start with and n = 200.
2. Because p = 0.80, take 0.755 – 0.80 = –0.045 (the numerator of the test
o
statistic).
3. Next, the standard error equals (the denominator
of the test statistic).
4. The test statistic is .
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