Question

Find mean and standard deviation in normal distribution example

Answer 1

Question :
Suppose that the ages of students in an intro to statistics class are normally distributed. We know that 5% of the students are older than 19.76 years. We also know that 10% of students are younger than 18.3 years.

What are the mean and standard deviation of the ages?

ANSWER :
Let's take the example in question. Assume that the mean is μ and that the standard deviation is σσ. If we have two z-values z1 and z2 corresponding to our two observations, 19.76 and 18.3 then we can solve the following equations for μ and σμ and σ.
19.76−μσ=z118.3−μσ=z2
We have two equations in two unknowns, solving which, we can find μ.
From your z-score table the data at 95% is at about mean +1.65 standard deviations. Taking μμ as the mean and σσ as the standard deviation, this tells us that μ+1.65σ=19.76 You should be able to write a similar equation from the other piece of data. That gives two equations in two unknowns.