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A manufacturer of space shuttle light bulbs claims that the defect rate of the bulbs is $0.1\%$. You suspect the defect rate is actually higher, so you have checked $1000$ identical light bulbs from this manufacturer and found out that 3 of them are defect.
Formulate the relevant hypotheses and test statistic (including its distribution) and investigate the claim of the manufacturer at significance level $α = 0.05$.

In my opinion, is a Binomial Distribution $Bin(1000,0.001)$.
For the hypothesis, it's correct to assume that $H_0:\mu=0.001$, $H_1:\mu>0.001$?
Then how can I use the significance level with a Binomial Distribution? (I know how to solve it but with a Normal) :(

For a $Bin(n,p)$ $P(X=k)=\frac{n!}{k!(n-k)!}\cdot p^k\cdot(1-p)^{n-k}$

TFAE
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    Why not apply the normal approximation to Binomial RVs via CLT. Seems appropriate here with high $N$ and small $p$. Anyway on the wikipedia page for Binomial RVs you can find many confidence intervals for the proportion $p$. https://en.m.wikipedia.org/wiki/Binomial_proportion_confidence_interval – Nap D. Lover Jan 23 '19 at 18:13
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    Won't work as the variance is too low. – Michael Hoppe Jan 23 '19 at 18:50
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    You may try Poisson instead with $\lambda=1$ and calculate $1-P(X\leq3)$ easily. – Michael Hoppe Jan 23 '19 at 18:56
  • @MichaelHoppe is not more accurate to use the binomial? – TFAE Jan 23 '19 at 20:12

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Slightly more accurate is $H_0:μ\le 0.001, H_1:μ>0.001$. That is, the manufacturer claims an upper limit. It doesn't matter for the analysis though. We will still use the same $p=0.001$.

We need to calculate $P(X\ge 3)$ to verify the hypothesis. If it is less than $\alpha=0.05$ we can reject the manufacturer's claim. We can find it with: $$P(X\ge 3)=1-P(X<3)=1-\big[P(X=0)+P(X=1)+P(X=2)\big]$$

Klaas van Aarsen
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