# Hypothesis testing.

Claim: $60\%$ of the light bulb will last more than $950$ hours.

To test the claim, $100$ bulbs were randomly tested and $68$ of them lasted more than $950$ hours. Can the claim be worthy vs $0.05$ level of significant?

I can't seems to get the $z$ value as I can't get the sample deviation. So how do we approach this problem ?

## 1 Answer

For one bulb, either it lasts longer than $950$ hours ("success"), or it doesn't ("failure"), i.e. for one bulb we have a Bernoulli random variable. The premise of the question is that the this is a Bernoulli RV with $p=0.6$.

If we draw multiple draws from a Bernoulli and count how many come out as "success" we have a RV with Binomial distribution. In this case we would have $B(100,0.6)$ since we are testing 100 bulbs and the assumed success probability is $0.6$. If you know a hypothesis test for Biomial RVs you're finished, otherwise we can take use a test for a normal RV.

For large sample sizes the binomial distribution $B(n,p)$ can be approximated by a normal distribution $N(np,\, np(1-p))$. Thus we can apply a hypothesis test for normal RVs to the normal distribution $N(60, \,24)$ at a confidence interval of $0.95$.