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# Hypothesis Testing of Proportions

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awson and Jick [1976] compare drug prescription in the United States and Scotland.
(a) In patients with congestive heart failure, two or more drugs were prescribed in 257 of 437 U.S. patients. In Scotland, 39 of 179 patients had two or more drugs prescribed. Test the null hypothesis of equal proportions giving the resulting p- value. Construct a 95% confidence interval for the difference in proportions.

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Biostatistics
Awson and Jick [1976] compare drug prescription in the United States and Scotland.
(a) In patients with congestive heart failure, two or more drugs were prescribed in 257 of 437 U.S. patients. In Scotland, 39 of 179 patients had two or more drugs prescribed. Test the null hypothesis of equal proportions giving the resulting p- value. Construct a 95% confidence interval for the difference in proportions.
The null hypothesis tested is
H0: There is no significant difference in the proportion of patients had two or more drugs prescribed in the United States and Scotland (P1 = P2)
The alternative hypothesis is
H1: There is significant difference in the proportion of patients had two or more drugs prescribed in the United States and Scotland (P1 â‰  P2)
The Test Statistic used is
where
Here p1 = 257/437 = 0.588100686, p2 = 39/179 = 0.217877095, n1 = 437, n2 = 179
Now = 0.480519481
Therefore, = 8.350276088
Rejection criteria: Reject the null hypothesis, if the observed significance (p-value) is less than the significance level 0.05.
P-value = P ( > 8.350276088) = 0
Conclusion: Reject the null hypothesis, since the observed significance (p-value) is less than the significance level 0.05. The sample provides enough evidence to conclude that there is significant difference in the proportion of patients had two or more drugs prescribed in the United States and Scotland.
Details
Z Test for Differences in Two Proportions

Data
Hypothesized Difference 0
Level of Significance 0.05
Group 1
Number of Successes 257
Sample Size 437
Group 2
Number of Successes 39
Sample Size 179

Intermediate Calculations
Group 1 Proportion 0.588100686
Group 2 Proportion 0.217877095
Difference in Two Proportions 0.370223592
Average Proportion 0.480519481
Z Test Statistic 8.350276088

Two-Tail Test
Lower Critical Value -1.959963985
Upper Critical Value 1.959963985
p-Value 0
Reject the null hypothesis

95% confidence interval is given by,
, where p1 = 257/437 = 0.588100686, p2 = 39/179 = 0.217877095, n1 = 437, n2 = 179, = 1.959963985
Therefore, required confidence interval is

= (0.294154929, 0.446292254)
Thus with 95% confidence we can claim that the difference in proportions is within (0.294, 0.446).
Details
Confidence Interval Estimate
of the Difference Between Two Proportions

Data
Confidence Level 95%

Intermediate Calculations
Z Value -1.959963985
Std. Error of the Diff. between two Proportions 0.038811255
Interval Half Width 0.076068662

Confidence Interval
Interval Lower Limit 0.294154929
Interval Upper Limit 0.446292254

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