# Each day a quality engineer selects a random sample of 60 power supplies

1. Apr 24, 2017

### Poke

1. The problem statement, all variables and given/known data
Each day a quality engineer selects a random sample of 60 power supplies from the day's production, measures their output voltages, and computes a 90% confidence interval for the mean output voltage of all the power supplies manufactured that day. What is the probability that more than 15 of the confidence intervals constructured in the next 210 days will fail to cover the true mean? Hint: Use the normal approximation.

2. Relevant equations
z=\Frac{x-\miu}{\sigma}

3. The attempt at a solution
First, X~Bin(n,p), as np >10, it follows Normal distribution, X~N(np, np(1-p))

np is \miu,
\sqrt{np(1-p)} is \signma

then Z = \frac{x-\miu}{\sigma}, it is the area of left tail

2. Apr 24, 2017

### Ray Vickson

If $X\sim\text{Bin}(n,p)$, the statement that $X$ "follows a normal distribution" when $np > 10$ is patently false: it may be approximately normal, at best. Furthermore, an $np$ of about 10 is too small for the normal to be really accurate, but using the "1/2" correction can improve the approximation quite a bit. Finally: looking at just $np$ is not good enough; you also need to look at $n(1-p)$; that is, you need both successes and failures to have moderate-to-large means.

Anyway, in this particular problem the normal approximation should be quite good. However, you have not shown any substantive calculations, so it is impossible to judge whether you know what to do next, and that is by far the most important issue.