AIMs: Define, calculate and interpret the mean and variance of the sample mean. Define and estimate the sample mean and sample variance. Define and construct a confidence interval.
Questions:
313.1. Defaults in a large bond portfolio follow a Poisson process where the expected number of defaults each month is four (λ = 4 per month). The number of defaults that occur during a single month is denoted by d(i). Therefore, over a one-year period, a sample of twelve observations is produced: d(1), d(2), ... , d(12). The average of these twelve observations is the monthly sample mean. This sample mean naturally has an expected value of four. Which is nearest to the standard error of this monthly sample mean; i.e., the standard deviation of the sampling distribution of the mean?
a. 0.11
b. 0.33
c. 0.58
d. 4.00
313.2. A random sample of 36 observations drawn from a normal population returns a sample mean of 18.0 with sample variance of 16.0. Our hypothesis is: the population mean is 15.0 with population variance of 10.0. Which are nearest, respectively, to the test statistics of the sample mean and sample variance (given the hypothesized values, naturally)?
a. t-stat of 3.0 and chi-square stat of 44.3
b. t-stat of 4.5 and chi-square stat of 56.0
c. t-stat of 6.8 and chi-square stat of 57.6
d. t-stat of 9.1 and chi-square stat of 86.4
313.3. A random sample of 41 hedge fund returns, drawn from a normal population, returns a sample mean of +5.0% with sample standard deviation of 2.0%. What are the two-sided 95% confidence intervals for, respectively, the population mean and population standard deviation (note: this requires non-calculator lookups or calculations, which are normally beyond the actual exam's scope)?
a. mean interval = {4.37%, 5.63%} and standard deviation interval = {1.64%, 2.56%}
b. mean interval = {4.05%, 5.93%} and standard deviation interval = {1.22%, 2.99%}
c. mean interval = {3.70%, 6.12%} and standard deviation interval = {0.94%, 3.18%}
d. mean interval = {3.22%, 7.49%} and standard deviation interval = {0.73%, 4.04%}
Answers:
Questions:
313.1. Defaults in a large bond portfolio follow a Poisson process where the expected number of defaults each month is four (λ = 4 per month). The number of defaults that occur during a single month is denoted by d(i). Therefore, over a one-year period, a sample of twelve observations is produced: d(1), d(2), ... , d(12). The average of these twelve observations is the monthly sample mean. This sample mean naturally has an expected value of four. Which is nearest to the standard error of this monthly sample mean; i.e., the standard deviation of the sampling distribution of the mean?
a. 0.11
b. 0.33
c. 0.58
d. 4.00
313.2. A random sample of 36 observations drawn from a normal population returns a sample mean of 18.0 with sample variance of 16.0. Our hypothesis is: the population mean is 15.0 with population variance of 10.0. Which are nearest, respectively, to the test statistics of the sample mean and sample variance (given the hypothesized values, naturally)?
a. t-stat of 3.0 and chi-square stat of 44.3
b. t-stat of 4.5 and chi-square stat of 56.0
c. t-stat of 6.8 and chi-square stat of 57.6
d. t-stat of 9.1 and chi-square stat of 86.4
313.3. A random sample of 41 hedge fund returns, drawn from a normal population, returns a sample mean of +5.0% with sample standard deviation of 2.0%. What are the two-sided 95% confidence intervals for, respectively, the population mean and population standard deviation (note: this requires non-calculator lookups or calculations, which are normally beyond the actual exam's scope)?
a. mean interval = {4.37%, 5.63%} and standard deviation interval = {1.64%, 2.56%}
b. mean interval = {4.05%, 5.93%} and standard deviation interval = {1.22%, 2.99%}
c. mean interval = {3.70%, 6.12%} and standard deviation interval = {0.94%, 3.18%}
d. mean interval = {3.22%, 7.49%} and standard deviation interval = {0.73%, 4.04%}
Answers:
