Chapter 9.4, Problem 55E

In medical investigations, the ratio θ = p₁/p₂ is often of more interest than the difference p₁ − p₂ (e.g., individuals given treatment 1 are how many times as likely to recover as those given treatment 2?). Let . $\hat{\theta }={\hat{p}}_1/{\hat{p}}_2$. When m and n are both large, the statistic ln $\left(\hat{\theta }\right)$ has approximately a normal distribution with approximate mean value ln(θ) and approximate standard deviation [(m − x)/(mx) + (n − y)/(ny)]^1/2.

a. Use these facts to obtain a large-sample 95% CI formula for estimating ln(θ), and then a CI for θ itself.

b. Return to the heart-attack data of Example 1.3, and calculate an interval of plausible values for θ at the 95% confidence level. What does this interval suggest about the efficacy of the aspirin treatment?