the rate of return for an investment can be described by a normal distribution with mean 22% and standard deviation 3%. what is the probability that the rate of return for the investment will be at least 15%?
To find the probability that the rate of return for the investment will be at least 15%, we can use the properties of the normal distribution. Here's how you can calculate it step by step:
Step 1: Standardize the value
To use the normal distribution table or calculator, you need to convert the value of 15% into a standardized z-score. The formula for standardizing a value is:
z = (x - μ) / σ
where z is the standardized value, x is the desired value (15% in this case), μ is the mean (22%), and σ is the standard deviation (3%).
Plugging in the values, we get:
z = (0.15 - 0.22) / 0.03
z = -0.07 / 0.03
z ≈ -2.33
Step 2: Find the corresponding probability
Once you have the standardized z-score, you can determine the probability associated with it. You can use a normal distribution table or a calculator to find this probability. In this case, we need to find the probability of being in the left tail of the distribution, i.e., the probability of obtaining a value that is less than or equal to -2.33.
Using a normal distribution table or calculator, the corresponding probability for a z-score of -2.33 is approximately 0.0099.
Step 3: Calculate the probability
Finally, to find the probability that the rate of return for the investment will be at least 15%, we subtract the probability we just found from 1, since we want the probability of the rate of return being greater than 15%.
Probability = 1 - 0.0099
Probability ≈ 0.9901
Therefore, the probability that the rate of return for the investment will be at least 15% is approximately 0.9901 or 99.01%.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to that Z score.