Question statement is like this: You brought a box of lightbulbs from a shop. You know that the bulbs are all either short-life bulbs with mean life of 500 hours or long-life bulbs with a mean life of 2500 hours, but you can’t tell which, because there was no label on the box.
As you have not shopped from this shop before, you initially have no opinion as to whether you have been sold long-life bulbs or the cheaper alternative.
After approximately 300 hours you find that 5 bulbs are alive. Assuming that life of an individual light bulb has an exponential distribution, how would you now assess the probability that you bought the long life bulbs?
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I dont understand the approach to solve it. I tried testing of hypothesis, but neither 'n' nor significance level is given. Tested using MP test, H0:2500 vs H1:500.
Another method I thought to try failure rate. but instead of failure rate, survival rate is given. I don't know if survival rate exist in stats ( 5 bulbs survived in 300 hrs).
Please help
The solution uses Bayes' Theorem. I'll try to get you started.
(1) Either you bought short-lived bulbs $S$ or long-lived bulbs $L$. Your prior probabilties (before testing the bulbs at home) is $P(S)=P(L) = 1/2.$
(2) Let $E$ be the event that five bulbs tested (presumably out of five) are still alive after 300 hours.
(3) You seek $P(L|E).$
Bayes' Theorem says $$P(L|E) = \frac{P(L\cap E)}{P(E)} = \frac{P(L)P(E|L)}{P(L)P(E|L)+P(S)P(E|S)}.$$
Knowing what you do about the exponential distribution, you should be able to find $P(E|L)$ and $P(E|S)$ in order to finish the problem.
What is the probability one 'long-lived' bulb lasts for at least 300 hours? What is the probability all of five long-lived bulbs last for at least 300 hours? And so on. with the short-lived bulbs?