# How to compute the confidence interval of the ratio of two normal means

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

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– mbq♦ Oct 2 '11 at 8:58

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@mbq - the Cauchy distribution presents no problems for confidence intervals, as the CDF is the inverse tangent function. Variance need not be defined for CIs to work. And the ratio of two normal RVs with zero mean is Cauchy, but not necessarily two normal RVs with non-zero mean. – probabilityislogic Oct 2 '11 at 10:58
@probabilityislogic Sure, I must stop trying to think on Sunday mornings. – mbq♦ Oct 2 '11 at 12:13

## 2 Answers

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, and an appendix to the first edition of my Intutitive Biostatistics is even more concise. I've created an online calculator that does the computation.

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It's very good references, I also like that you actually made a calculator for it (+1). As expected though, in your calculator you clearly state that when the confidence interval of the denominator includes zero, it is not possible to compute the CI of the quotient. I think it's the same that happens when I try solving the quadratic equation. suppose variance is 1, mu1 = 0 and mu2=1, N=10000. It's unsolvable. – francogrex Oct 2 '11 at 16:03

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thanks for the online calculator Harvey, I'm a typical biologist with insufficient background in statistics and your calculator was exactly what I needed. – Timtico Jul 5 '12 at 8:29

R has the package `mratios` with the function `t.test.ratio`.

Gemechis Dilba Djira, Mario Hasler, Daniel Gerhard and Frank Schaarschmidt (2011). mratios: Inferences for ratios of coefficients in the general linear model. R package version 1.3.15. http://CRAN.R-project.org/package=mratios

See also http://www.r-project.org/user-2006/Slides/DilbaEtAl.pdf

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