How to find Standard Deviation, given Mean and Cumulative Normal Distribution?

Question

The question specifically:

If X ∼ N(20, σ2) and Pr(X ≥ 19) = 0.7, find the standard deviation, σ.

I just don't quite understand how I can find the SD here?

I assume I need to find variance, and as such an expected value, but how might I go about this without a table of values?

Answer 1

Following @StubbornAtom's Comment of a couple of days ago, this suggests how to find the answer. Make sure you can use methods and tables in your text to get the answer.

To start: $$P(X \ge 19) = P\left(\frac{X-\mu}{\sigma} \ge \frac{19-20}{\sigma} = -\frac 1 \sigma\right) = P(Z \ge -1/\sigma) = .07.$$

But from standard normal tables or from software you know that $P(Z \ge -0.5255) = 0.7,$ whers $Z \sim \mathsf{Norm}(0,1).$ Do you know how to use a printed table to get close to this result?

Then finally, how do you find $\sigma?$

Computations in R statistical software:

qnorm(.3)
[1] -0.5244005            # c = -0.5244 has P(Z < c) = 0.3 
1 - pnorm(19, 20, 1.907)
[1] 0.6999942             # P(X > 19) = 0.7 if X ~ NORM(mu=20, sigma=1.907)