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We now consider the case where there are two binomial parameters of interest, say, p1 and p2, and we wish to draw inferences about these proportions. We will present large-sample hypothesis testing and con dence interval procedures based on the normal approximation to the binomial.
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Large-Sample Test for H0: p1
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Suppose that two independent random samples of sizes n1 and n2 are taken from two populations, and let X1 and X2 represent the number of observations that belong to the class of interest in samples 1 and 2, respectively. Furthermore, suppose that the normal approximation to the binomial is applied to each population, so the estimators of the population proportions
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P1 X1 n1 and P2 X2 n2 have approximate normal distributions. We are interested in testing the hypotheses
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H0: p1 H1: p1 The statistic B
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is distributed approximately as standard normal and is the basis of a test for H0: p1 p2. Speci cally, if the null hypothesis H0: p1 p2 is true, using the fact that p1 p2 p, the random variable Z B
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1 p2 a n
1 n2 b
is distributed approximately N(0, 1). An estimator of the common parameter p is
X1 n1
X2 n2
The test statistic for H0: p1
p2 is then Z0
P1 P 11
1 P2 a n
1 n2 b
This leads to the test procedures described below.
Null hypothesis: Test statistic:
H0: p1 Z0
P1 P 11 P2 1 P2 a
1 n2 b Rejection Criterion z0 z0 z0 z z
2 or z0
Alternative Hypotheses H1: p1 H1: p1 H1: p1 p2 p2 p2
Extracts of St. John s Wort are widely used to treat depression. An article in the April 18, 2001 issue of the Journal of the American Medical Association ( Effectiveness of St. John s Wort on Major Depression: A Randomized Controlled Trial ) compared the ef cacy of a standard extract of St. John s Wort with a placebo in 200 outpatients diagnosed with major depression. Patients were randomly assigned to two groups; one group received the St. John s Wort, and the other received the placebo. After eight weeks, 19 of the placebo-treated patients showed improvement, whereas 27 of those treated with St. John s Wort improved. Is there any reason to believe that St. John s Wort is effective in treating major depression Use 0.05. The eight-step hypothesis testing procedure leads to the following results: The parameters of interest are p1 and p2, the proportion of patients who improve following treatment with St. John s Wort ( p1) or the placebo ( p2). 2. H0: p1 p2 3. H1: p1 p2 4. 0.05 5. The test statistic is 1. z0 B
p1 p 11 p2
1 p2 a n 19 100 19 100
1 n2 b 0.19, n1 n2 100, and
where p1
27 100
0.27, p2
x1 n1
x2 n2
27 100
0.23 z0.025 1.96.
6. 7.
p2 if z0 z0.025 1.96 or if z0 Computations: The value of the test statistic is z0 1 0.2310.772 a 100 B 0.27 0.19 1 b 100
Reject H0: p1
Conclusions: Since z0 1.35 does not exceed z0.025, we cannot reject the null hypothesis. Note that the P-value is P 0.177. There is insuf cient evidence to support the claim that St. John s Wort is effective in treating major depression.
The following box shows the Minitab two-sample hypothesis test and CI procedure for proportions. Notice that the 95% CI on p1 p2 includes zero. The equation for constructing the CI will be given in Section 10-6.4. Test and CI for Two Proportions Sample X N Sample p 1 27 100 0.270000 2 19 100 0.190000 Estimate for p(1) p(2): 0.08 95% CI for p(1) p(2): ( 0.0361186, 0.196119) Test for p(1) p(2) 0 (vs not 0): Z 1.35 P-Value 0.177
364 10-6.2 10-6.3
Small-Sample Test for H0: p1
p2 (CD Only)
-Error and Choice of Sample Size
The computation of the -error for the large-sample test of H0: p1 p2 is somewhat more involved than in the single-sample case. The problem is that the denominator of the test statistic Z0 is an estimate of the standard deviation of P1 P2 under the assumption that p1 p2 p. When H0: p1 p2 is false, the standard deviation of P1 P2 is B p1 11 p1 2 p2 11 p2 2