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Comparing two proportions, like comparing two means, is common. If two estimated proportions are different, it may be due to a difference in the populationsor it may be due to chance. A hypothesis test can help determine if a difference in the estimated proportions reflects a difference in the population proportions.
The difference of two proportions follows an approximate normal distribution. Generally, the null hypothesis states that the two proportions are the same. That is, . To conduct the test, we use a pooled proportion, .
Two types of medication for hives are being tested to determine if there is a difference in the proportions of adult patient reactions. Twenty out of a random sample of 200 adults given medication A still had hives 30 minutes after taking the medication. Twelve out of another random sample of 200 adults given medication B still had hives 30 minutes after taking the medication. Test at a 1% level of significance.
This is a test of 2 population proportions.
How do you know?
The problem asks for a difference in proportions.
Let and be the subscripts for medication A and medication B. Then and are the desired population proportions.
The words "is a difference" tell you the test is two-tailed.
Distribution for the test: Since this is a test of two binomial population proportions, the distribution is normal:
Therefore,
follows an approximate normal distribution.
Calculate the p-value using the normal distribution: p-value = 0.1404.
Estimated proportion for group A:
Estimated proportion for group B:
.
Half the p-value is below -0.04 andhalf is above 0.04.
Compare and the p-value: and the . p-value.
Make a decision: Since , do not reject .
Conclusion: At a 1% level of significance, from the sample data, there is not sufficient evidence to conclude that there is a difference in the proportions of adultpatients who did not react after 30 minutes to medication A and medication B.
STAT
. Arrow over to
TESTS
and press
6:2-PropZTest
. Arrow down and enter
20
for
,
200
for
,
12
for
,
and
200
for
. Arrow down to
p1
: and arrow to
not equal p2
. Press
ENTER
. Arrow down to
Calculate
and press
ENTER
. The p-value is
and the test statistic is 1.47. Do the procedure again but instead
of
Calculate
do
Draw
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