Confidence Interval – How to Compute for Ratio of Two Normal Means

confidence intervalmeannormal distributionratio

I want to derive the limits for the $100(1-\alpha)\%$ confidence interval for the ratio of two means.
Suppose, $X_1 \sim N(\theta_1, \sigma^2)$ and $X_2 \sim N(\theta_2, \sigma^2)$
being independent, the mean ratio $\Gamma = \theta_1/\theta_2$. I tried to solve:
$$\text{Pr}(-z(\alpha/2)) \leq X_1 – \Gamma X_2 / \sigma \sqrt {1 + \gamma^2} \leq z(\alpha/2)) = 1 – \alpha$$ but that equation couldn't be solved for many cases (no roots). Am I doing something wrong? Is there a better approach? Thanks

Best Answer

Fieller's method does what you want -- compute a confidence interval for the quotient of two means, both assumed to be sampled from Gaussian distributions.

  • The original citation is: Fieller EC: The biological standardization of Insulin. Suppl to J R Statist Soc 1940, 7:1-64.

  • The Wikipedia article does a good job of summarizing.

  • I've created an online calculator that does the computation.

  • Here is a page summarizing the math from the first edition of my Intuitive Biostatistics

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