Normal distribution

Question

I have this question:

A normal distribution is such that 16% of it is smaller than 13, and 2.5% of it is larger than 22. What's the mean of this normal distribution?

I know I should be using the 68-95-99.7 rule, but I have no clue how to start without the standard deviation being given. Any hints are appreciated, thank you.

Answer 1

Hint:

• You are supposed to come up with two equations, relating the unknown mean and unknown standard deviation to $13$ in the first case and $22$ in the second

• Then solve these as simultaneous equations in two unknowns to find the mean; you could also find the standard deviation.

Answer 2

HINT: $\pm$ one standard deviation leaves $16\%$ above and $16\%$ below. Can you come up with something similar for $\pm$ two standard deviations?