To find the variance of the returns over the past five years, follow these detailed steps:
1. Calculate the Mean of the Returns:
- List of returns over five years: 17%, 13%, 8%, 2%, and 5%.
- Convert these percentages to decimal form for calculations: 0.17, 0.13, 0.08, 0.02, 0.05.
- Calculate the mean (average):
[tex]\[
\text{Mean} = \frac{0.17 + 0.13 + 0.08 + 0.02 + 0.05}{5} = \frac{0.45}{5} = 0.09
\][/tex]
2. Calculate the Squared Differences from the Mean:
- For each return, subtract the mean and then square the result:
[tex]\[
(0.17 - 0.09)^2 = (0.08)^2 = 0.0064
\][/tex]
[tex]\[
(0.13 - 0.09)^2 = (0.04)^2 = 0.0016
\][/tex]
[tex]\[
(0.08 - 0.09)^2 = (-0.01)^2 = 0.0001
\][/tex]
[tex]\[
(0.02 - 0.09)^2 = (-0.07)^2 = 0.0049
\][/tex]
[tex]\[
(0.05 - 0.09)^2 = (-0.04)^2 = 0.0016
\][/tex]
3. Calculate the Variance:
- Sum all the squared differences:
[tex]\[
0.0064 + 0.0016 + 0.0001 + 0.0049 + 0.0016 = 0.0146
\][/tex]
- Divide this sum by the number of returns (5, as we're dealing with a population here):
[tex]\[
\text{Variance} = \frac{0.0146}{5} = 0.00292
\][/tex]
4. Round the Variance to the Nearest Hundredth (if necessary):
- The variance calculated is 0.00292, which remains 0.0029 when rounded to the nearest hundredth.
Hence, the variance of the returns is approximately 0.0029, which, in this context, is not a direct match with any of the options provided. This variance is much smaller compared to the given choices. The options given (18.74, 29.2, 5.4, 13.88) are significantly different from our correct calculation indicating a potential misunderstanding in the provided choices or a possible typographical error in the question.
Therefore, the correct calculation of the variance from the data provided is 0.0029.
