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A manufacturer of a certain component determines that on average only $2$ components fail before a $1000$ hours of operation. A buyer detects that they fail $5$ before the $1000$ hours. If the number of failing components is a Poisson random variable, is there sufficient evidence to doubt the manufacturer?

Just in the previous activity I was asked to graph the probability functions of $Y$~$P (2)$ and $Z$~$P (5)$. And I thought that the $Y$ could represent the manufacturer and the $Z$ the buyer:

  • $$f_Y$$ Y~P(2)

  • $$f_Z$$ Z~P(5)

Then calculate $F_Y (5)$ which gave me $0.99$ and $F_Z (5)$ which gave me $0.62$.

This means that according to the manufacturer there is a ~ $99$% failure of up to $5$ components before the $1000$ hours and that according to the buyer there is ~ $62$%?

If the above is correct, then the answer is that there is not enough evidence to doubt the manufacturer, but quite the opposite?

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    You might also try posting on [Cross Validated](https://stats.stackexchange.com) – 2017-02-08

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I would say there isn't extremely strong evidence to doubt the manufacturer, but not that there is 'far from enough' evidence.

You want the probability, according to the manufacturer, that five or more fail. Since the manufacturer says that the failures should be approximately Poisson-distributed with mean $2$, the probability that there are five or more failures is $$P(N\ge 5) = \sum_{k=5}^\infty e^{-2}\frac{2^k}{k!} = 1 - e^{-2}\left(\frac{2^0}{0!}+\frac{2^1}{1!}+\frac{2^2}{2!}+\frac{2^3}{3!}+\frac{2^4}{4!}\right) = 1-7e^{-2}\approx 5.27\% $$

That means that if the manufacturer is right, there is only a $5.3\%$ chance that you would have this bad of an experience. The customary 'threshold for statistical significance' is below $5\%$ so this is right near that margin.