Here's the question that I'm having a problem with:
After reviewing previous loan records, the credit manager of a bank determines
that the data follows a normal distribution. The debts have a mean of $20000
and the probability that the loss could be greater than $25000 or less than
$15000 is 0.418. Determine the standard deviation of the data to the nearest
hundred dollars.
After looking at the answer (shown below), I can't understand how they're able to figure out to do $\frac{0.418}{2}$. I understand that they're using inversion on $\frac{0.418}{2}$ to get $-0.81$. Can someone please explain it better and maybe include a normal distribution curve to visualize it? I can't see where $0.418$ is on the normal distribution curve...
$$A_{(Z_1)} = \frac{0.418}{2} = 0.2090$$
$$Z_1 = -0.81$$ $$-0.81 = \frac{15000 - 20000}{\sigma} $$ $$-0.81 = \frac{5000}{\sigma} $$ $$-0.81\sigma = 5000$$ $$\frac{-0.81\sigma}{-0.81} = \frac{5000}{-0.81}$$ $$\sigma = 6172.8395 \approx 6200 $$
The events they are concidering are the events in the tails of the distribution, i.e. less than some number at the left side and larger than some number at the right. Since the normal distribution is symmetric and they are concidering symmetric deviations around the mean, you know that, for example, the probability of a loss less than 15000 is $0.418/2$. Then you look at the CDF of the standard normal distribution and find that -0.81 standard deviations has the probability $0.418/2$ (the deviation is negative since we are at the left of the mean). So if $X$ is the loss and $\Phi(Z)$ the CDF of the standard normal distribution, we have $$ P(X \leqslant 15000) = \Phi \left(\frac{15000 - 20000}{\sigma} \right) = \frac{0.418}{2} \Leftrightarrow \\ \frac{-5000}{\sigma} = -0.81 \\ \sigma = \frac{5000}{0.81}$$