An investment broker reports that the yearly returns on common stocks are approximately normally distributed with a mean return of 12.4 percent and a standard deviation of 20.6 percent. On the other hand, the firm reports that the yearly returns on tax-free municipal bonds are approximately distributed with a mean 5.2 percent and a standard deviation of 8.6 percent. Find the probability that a randomly selected
a. Common stock will give a positive yearly return.
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To find the probability that a randomly selected common stock will give a positive yearly return, we need to calculate the area under the normal distribution curve to the right of zero.
Since the mean return of common stocks is 12.4 percent and the standard deviation is 20.6 percent, we can use these values to calculate the probability.
Step 1: Standardize the value
To standardize the value, we need to convert it to a z-score using the formula:
z = (x - μ) / σ
Where:
x = the value we are interested in (zero in this case)
μ = the mean return of common stocks (12.4 percent)
σ = the standard deviation of common stocks (20.6 percent)
Plugging in the values, we get:
z = (0 - 12.4) / 20.6 = -0.6019
Step 2: Find the probability
Once we have the z-score, we can use a standard normal distribution table or a calculator to find the probability associated with that z-score.
Looking up the z-score of -0.6019 in the standard normal distribution table, we find that the probability is 0.2743.
Step 3: Interpret the result
So, the probability that a randomly selected common stock will give a positive yearly return is approximately 0.2743, or 27.43%.
Note: This calculation assumes that the returns on common stocks follow a perfectly normal distribution, which may not be exactly the case in reality.