Find the nth percentile

Question

I am given the β=4... Meaning this is the mean and SD

But I have to find the tenth percentile. I tried entering this into a software and found Z. Which is -1.282 and entered it into a formula of xo = mean + (Z)(SD).

But my answer was wrong.. what approach should I do?

Answer 1

It seems you have $X \sim \mathsf{Norm}(\mu = 4,\, \sigma = 4).$ You seek $x$ such that $P(X \le x) = 0.10.$

If you are doing this using printed tables of the standard normal CDF, then the following approach would work:

$$0.10 = P(X \le x) = P\left(\frac{X - \mu}{\sigma} \le \frac {x - 4}{4}\right) = P\left(Z \le \frac{x-4}{4}\right).$$

From the table you should be able to get $(x - 4)/4 \approx -1.2816.$ then $x \approx -1.1262.$ (Different printed tables have different formats, so I can't tell you exactly how to use the table you happen to be using.) Depending on how much rounding you have to do to use your printed table, your answer may be a little different.

Some kinds of statistical software and statistical calculators allow you to make this computation without standardizing. For example, in R statistical software qnorm is the quantile (or inverse-CDF) function of a normal distribution with specified parameters:

qnorm(.1, 4, 4)
## -1.126206