A Swedish study used 1360 patients who had suffered a stroke to determine if an

Question

A Swedish study used 1360 patients who had suffered a stroke to determine if an aspirin treatment reduced the risk of dying from a heart attack compared to a placebo. The study randomly assigned 684 subjects to the placebo group and 676 subjects to the aspirin group. In the placebo group 28 subjects died from a heart attack, whereas 18 in the aspirin group died from a heart attack. At the 5% significance level, does the aspirin treatment reduce the risk of dying from a heart attack compared to the placebo treatment? Give the corresponding 95% confidence interval for the parameter of interest.

Explanation / Answer

Data:    

n1 = 676   

n2 = 684   

p1 = 18/676 = 0.026627219   

p2 = 28/684 = 0.040935673   

Hypotheses:    

Ho: p1 p2    

Ha: p1 < p2    

Decision Rule:    

= 0.05   

Lower Critical z- score = -1.644853627   

Reject Ho if z < -1.644853627   

Test Statistic:    

Average proportion, p = (n1p1 + n2p2)/(n1 + n2) = (676 * 0.0266272189349112 + 684 * 0.0409356725146199)/(676 + 684) = 0.033823529

q = 1 - p = 1 - 0.0338235294117647 = 0.966176471

SE = [pq * {(1/n1) + (1/n2)}] = (0.0338235294117647 * 0.966176470588235 * ((1/676) + (1/684))) = 0.009804055

z = (p1 - p2)/SE = (0.0266272189349112 - 0.0409356725146199)/0.00980405514653048 = -1.45944238

p- value = 0.072221694   

Decision (in terms of the hypotheses):    

Since -1.45944238 > -1.644853627 we fail to reject Ho

Conclusion (in terms of the problem):    

There is no sufficient evidence that p1 < p2

n1 = 676

n2 = 684

p1 = 0.026627219

p2 = 0.040935673

% = 95

Pooled Proportion, p = (n1 p1 + n2 p2)/(n1 + n2) = (676 * 0.0266272189349112 + 684 * 0.0409356725146199)/(676 + 684) = 0.033823529

q = 1 - p = 1 - 0.0338235294117647 = 0.966176471

SE = (pq * ((1/n1) + (1/n2))) = (0.0338235294117647 * 0.966176470588235 * ((1/676) + (1/684))) = 0.009804055

z- score = 1.959963985

Width of the confidence interval = z * SE = 1.95996398454005 * 0.00980405514653048 = 0.019215595

Lower Limit of the confidence interval = (p1 - p2) - width = -0.0143084535797086 - 0.0192155949896443 = -0.033524049

Upper Limit of the confidence interval = (p1 - p2) + width = -0.0143084535797086 + 0.0192155949896443 = 0.004907141

The 95% confidence interval is [-0.0335, 0.0049]

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