# Normal distribution problem with expectation and deviation values not known.

I am trying to solve the following problem.

The diameter of a product manufactured can be regarded as a random variable E. During an inspection products with a thickness greater than 1.01 mm or less than 0.99 mm are sorted out. It has been found out that:

P(E > 1.01) = 8%

P(E < 0.99) = 2%

determine the mean and standard deviation.

What i have concluded is the following:

P(0.99 < E < 1.01) = 1 - (0.08 + 0.02) = 0.90

So I suppose that I need to do some reverse lookup in the normal distribution table but what I can see I have two unknown variables, both the mean and the deviation. I think I need some help to proceed.

• Combining the two probabilities is not useful, it loses information. Let the mean be $\mu$ and the standard deviation be $\sigma$. The right tail of the standard normal has area $0.08$ if you are at about $1.405$ (reverse look up). So $\mu+1.405\sigma=1.01$. Get a similar equation for the other tail, and solve for $\mu$ and $\sigma$. – André Nicolas Oct 25 '15 at 17:35

## 1 Answer

In your table, what is the length between the 2 values corresponding to these probabilities ? Since in your real case it is 0.02, it tells you how to scale the variable... and thus the standard deviation. From here I guess you will know how to solve the last degree of freedom, right ? :-)