Estimating the Expectation

A measurement follows the Normal distribution with standard deviation that is equal to 15 and an unknown expectation μ. Two statisticians propose two distinct ways to estimate the unknown quantity μ with the aid of a sample of size 36. Statistician A proposes to use the sample average as an estimate. Statistician B proposes to use the sample median instead. In order to choose between the two options they agree to prefer the statistic that has a smaller variance (with respect to the sampling distribution). Tasks 1-9 refer to this problem of comparing the two statistics to each other.

Question 1

Assume that the actual expectation of the measurement is equal to 5 (μ=5). Then the expectation of the statistic that was proposed by Statistician A is equal to:
Answer:

Question 2
Assume that the actual expectation of the measurement is equal to 5 (μ=5). Then the standard deviation of the statistic that was proposed by Statistician A is equal to:
Answer:

Question 3
Assume that the actual expectation of the measurement is equal to 5 (μ=5). Then the expectation of the statistic that was proposed by Statistician B is equal to: (In order to answer this question you may need to conduct a simulation, similar to the simulations that are presented in the book. Choose the option closest to the outcome of the simulation.)
Choose one answer.

a. 2.3

b. 3.7

c. 4.1

d. 5.0

Question 4
Assume that the actual expectation of the measurement is equal to 5 (μ=5). Then the standard deviation of the statistic that was proposed by Statistician B is equal to: (In order to answer this question you may need to conduct a simulation, similar to the simulations that are presented in the book. Choose the option closest to the outcome of the simulation.)
Choose one answer.

a. 2.0

b. 2.5

c. 3.0

d. 3.5

Question 5
Assume that the actual expectation of the measurement is equal to 2.3 (μ=2.3). Then the expectation of the statistic that was proposed by Statistician A is equal to:
Answer:

Question 6
Assume that the actual expectation of the measurement is equal to 2.3 (μ=2.3). Then the standard deviation of the statistic that was proposed by Statistician A is equal to:
Answer:

Question 7
Assume that the actual expectation of the measurement is equal to 2.3 (μ=2.3). Then the expectation of the statistic that was proposed by Statistician B is equal to: (In order to answer this question you may need to conduct a simulation, similar to the simulations that are presented in the book. Choose the option closest to the outcome of the simulation.)
Choose one answer.

a. 2.3

b. 3.7

c. 4.1

d. 5.0

Question 8
Assume that the actual expectation of the measurement is equal to 2.3 (μ=2.3). Then the standard deviation of the statistic that was proposed by Statistician B is equal to: (In order to answer this question you may need to conduct a simulation, similar to the simulations that are presented in the book. Choose the option closest to the outcome of the simulation.)

Choose one answer.

a. 2.0

b. 2.5

c. 3.0

d. 3.5

Question 9

Based on the information collected in Tasks 1-8, which of the two statistics produces values which tends to be more concentrated about the expectation of the measurement?

Choose one answer.

a. The statistic proposed by Statistician A

b. The statistic proposed by Statistician B
Normal Approximation of the Sampling Distribution of a Sum

Suppose that the expected number of phone calls that are handel by a switchboard in each second is 5.35. Assume that the distribution of the number of phone calls per second follows the Poisson distribution. Tasks 10-12 refer to this information.

Question 10

The number of phone calls that the switchboard is expected to handle in 1 minute is

Question 11

The 0.80-percentile of the number of calls per minute is (approximately) equal to: (Use the Normal approximation. The answer may be rounded up to 3 decimal places of the actual value.)

Question 12

The probability that the switchboard will need to handel no more than 300 call in a minute is: (Use the Normal approximation, without a continuity correction. The answer may be rounded up to 3 decimal places of the actual value.)