Difference in recovery rates between two treatments

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In a trial, treatment X recovers 45 of 150 patients and treatment Y recovers 60 of 150. Construct a 95% confidence interval for the difference in recovery rates (Y minus X), and state your conclusion.

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Setup. Use (\hat{p}Y - \hat{p}X) \pm z^{}\sqrt{\hat{p}X(1-\hat{p}X)/nX + \hat{p}Y(1-\hat{p}Y)/nY} with z^{} = 1.96. Here \hat{p}X = 0.30, \hat{p}Y = 0.40, difference = 0.10. Compute the standard error. \sqrt{\frac{0.30 	imes 0.70}{150} + \frac{0.40 	imes 0.60}{150}} = \sqrt{0.0014 + 0.0016} = \sqrt{0.003} = 0.0548. Margin of error = 1.96 	imes 0.0548 = 0.1074, so 0.10 \pm 0.1074 = [-0.007,\ 0.207]. Interpret. The interval includes zero, so at the 5% level the recovery rates are not significantly different, even though Y looks 10 points better. The sample of 150 per arm is too small to conf…

Difference in recovery rates between two treatments

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