Normal distribution about probability of stocks revenue (positive and negative)

Question

So I am struggling with a question in statistics following :

An investment broker reports that the yearly returns on common stocks are approximately normally distributed with a mean of 12,4% and a standard deviation of 20,6%. On the other hand, the firm reports that the yearly returns on tax-free municipal bonds are approximately normally distributed with a mean return of 5,2% and a standard deviation of 8,6% Find the probability that a randomly yearly selected :

c) Common stock will give more than a 10% return.

Formula : $P(0,01>b)=1-P(X\le b)$

$Z=0,10-12,40/20,60 = 0,597$

Which gives me 0,2743 on normal table and :

$1-0,2743 = 0,7257$

So this should in theory mean the probability of more than 10% revenue is 0,7257?

The next question is with a loss of at least 10% how should I compute that?

Answer 1

Using software to check first computation: You have $X \sim \mathsf{Norm}(\mu=12.4, \sigma=20.6).$ You seek $P(X > 10) = 1 - P(X \le 10) = 0.5464.$

Computations are in R statistical software, where pnorm is a normal CDF. Because of rounding involved in the use of normal tables, your answer might be a little different. [It seems you have the right approach, but with a mistake at the start. I will leave it to you to fix the standardization.]

1 - pnorm(10, 12.4, 20.6)
## 0.5463738

For the next part you seek $$P(X < -10) = P\left(\frac{X-\mu}{\sigma} < \frac{-10-12.4}{20.6}\right) = P(Z < -1.087) = \dots \approx 0.1384.$$

pnorm(-10, 12.4, 20.6)
##  0.1384348

The figure below shows the density function of $\mathsf{Norm}(12.4, 20.6),$ with vertical lines at $\pm 10.$ The first answer is the area under the normal curve to the right of the right-hand vertical line; the second is the area under the curve to the left of the left-hand line.

Note: It is always a good idea to make sketches of normal curves and areas when working problems like these. By hand, you can't make a sketch as accurate as the figure above, but with some practice you can make a sketch that is accurate enough to catch gross computational errors. (For example, there is no way the area to the right of the vertical blue line exceeds 70%.)