Question: A random sample of 85 group leaders, supervisors, and similar personnel revealed that on average a person spent 6.5 years on the job before being promoted. The standard deviation of the sample was 1.7 years.
Using the 0.95 degree of confidence, what is the confidence interval within which the population mean lies?
Select an Answer: 4.15 and 7.15 6.99 and 7.99 6.49 and 7.49 6.14 and 6.86
The interval estimate can be found from:
Here, we have n = 85, X̄ = 6.5, z = 1.96 (for 95%), and s = 1.7.
Therefore, 6.5 ± 1.96 × 1.7/9.22 and we get 6.14 and 6.86.
Question: The correlation coefficient between two assets is 1, the standard deviation on asset 1 is 10, and the standard deviation on asset 2 is 20.
What is the standard deviation on a portfolio in which both asset 1 and asset 2 each account for 50% of portfolio value?
Select an Answer: 206 15 13.2
The covariance is equal to the correlation coefficient multiplied by the two standard deviations. In this case, the covariance equals:
1 × 10 × 20 = 200
The standard deviation of a portfolio with two assets is equal to the square root of the following:
weight of asset 1 (w1) squared multiplied by the standard deviation of asset 1 (σ1) squared, plus the weight of asset 2 (w2) squared multiplied by the standard deviation of asset 2 (σ2) squared, plus two times the weight of asset 1 multiplied by the weight of asset 2 multiplied by the covariance (cov12). That is:
In this example, the standard deviation of the portfolio is equal to:
Population standard deviation is the square root of the population variance = 3.958.
x − Mean
(x − Mean)2
Question: Neal Jung, a quantitative analyst with HarvestTime Brokerage, is examining a data sample and has amassed the following information:
Standard deviation of the sample: 70
Number of observations: 600
Sample mean: 812
Assume that Mr. Jung formulates a null hypothesis that states that the value of the population mean is equal to 800. Additionally, assume that the population standard deviation is unknown.
Given this information, what is the standard error of the estimate? Further, what is the test statistic?
Select an Answer: 8.370; 4.148 0.014; 11.834
none of these answers 8.370; 1.434 0.014; 857.143
The standard error and test statistic for this example is 2.858 and 4.199, respectively. Therefore, none of these answers is correct.
If the population standard deviation is unknown, as in this example, the standard error of the estimate is found by using the following equation:
where s = the sample standard deviation and n = the number of observations in the sample.
In this example, all of the necessary information has been provided, and the determination of the standard error of the estimate is found as:
Standard error = 70/24.495 = 2.858
Now that the standard error of the estimate has been calculated, the test statistic can be found by using the following equation:
Test statistic = (Sample statistic − Value of the population parameter under the null hypothesis) / Standard error of the sample statistic
Again, all of the necessary information has been provided, and the calculation of the test statistic is found as follows:
Test statistic = (812 − 800) / 2.858 = 4.199
Question: You are given assets X, Y, and Z, which have expected returns of 5%, 10%, and 15% respectively, and standard deviations of return of 5%, 10%, and 15% respectively. Your client views any return below a level of 0% as unacceptable.
Find the asset that minimizes the probability that the portfolio will fall below 0% annual return, and what is the probability?
Select an Answer: Z, 14% none of these answers X, 15% Y, 17%
To answer this question we must follow Roy's safety-first criterion. First, we must find the shortfall level, R1. Fortunately, this is already given to us in the question, and it is 0%. Second, we compute the SFRatio for each asset in the portfolio:
Next, find the standard normal cdf evaluated at the SFRatio for each asset. The probability of shortfall will be N(-SFRatio). For X, Y, and Z, these will be N(-1), N(-1), and N(-1). Since N(-1) = 1 − N(1), and so on, we get 0.1587 for all three.
Thus, all assets equally minimize the probability that the return will fall short of 0, with a probability of approximately 16%.