Question

Dr. David Banner wishes to estimate the proportion of gamma ray machines that malfunction and produce excess radiation. A random sample of 65 gamma ray machines is selected. 15 of those machines malfunction. Compute the 95% confidence interval on the proportion of gamma ray machines that malfunction in the population.

Answer 1

You have to make an assumption about the prior distribution of failure rates for gamma ray machines in general. It is usually fairly safe to say "let's assume a prior distribution of failure rates as uniformly distributed between 0 and 100%". With this assumption, you can model the population failure rate as the 95% confidence interval of a Beta-Binomial distribution with parameters 50(number of observed successes)+1 and 15(number of observed failures)+1.

Answer 2

In Excel, I find the 2.5th percentile using =BETAINV(0.025,1+65-15,1+15) and the 97.5th percentile using =BETAINV(0.975,1+65-15,1+15).

Answer 3

Oh, sorry, I am modeling the success rate, so just flip that around for the failure rate.

Answer 4

thank you!!

Answer 5

Sure. You can also employ the Normal approximation here (which is fine, but not best, because np (65*.23) and n(1-p) (65*.77) are greater than 5.) In that case it looks like:
\[p\pm Z_{1-\alpha/2}\sqrt{\frac{p(1-p)}{n}}\]
where alpha is 5%, p is 15/65 and n is 65. (Z is the Normal distribution Z score).

Answer 6

okay thanks this is confusing to me but you explain it very well i appreciate it!