# Hypothesis testing for mean

I have a system with two components, component A and component B. Let $X$ be the random variable representing time between failures of component A and $X$ is exponentially distributed. Let $E[X]=m$. Now we want to test the following hypothesis:

null hypothesis: $m\le m_0$ where $m_0$ is some given constant alternate hypothesis: $m>m_0$

To test the hypothesis we need samples of $X$ which are not available. But, we have the information that A has a failure rate lower than B (time between failures for B is also exponentially distributed). MTTF of component B is $n_0$. Can we use samples generated from B and use as samples of A to test the above hypothesis?

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You know that the failure rate $h_A$ of A is smaller than the failure rate $h_B$ of B. Since the failures of B are exponentially distributed with mean $n_0$, we have that $h_B=1/n_0$. Since X is also exponentially distributed,$$h_A=\frac{1}{E[X]}=\frac{1}{m}<h_B=\frac{1}{n_0},$$ that is, $m>n_0$. So if $n_0$ is a known quantity, then the null hypothesis $m\leq m_0$ is trivially false if $m_0\leq n_0$ and untestable if $n_0<m_0$ since samples of B have no information about A. If $n_0$ is not known, test null hypothesis $n_0\leq m_0$ and if you reject it, then reject $m\leq m_0$ also. –  Dilip Sarwate Nov 10 '11 at 20:17