Question

A measurement follows the normal distribution with a standard deviation of 15 and an unknown expectation μ. You can consider that measurement to be the "original" distribution. Two statisticians propose two distinct ways to estimate the unknown quantity μ with the aid of a sample of size 36. They will do that by evaluating two different SAMPLING DISTRIBUTIONS to determine which method is better. There are two statisticians involved in this task: statistician "A" and statistician "B." Statistician A proposes to use the sampling distribution of the sample average as an estimate. Statistician B proposes to use the sampling distribution of 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.

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: ____________

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: _______

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: ________Select one: a. 2.3 b. 3.7 c. 4.1 d. 5.0

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: Select one: a. 2.0 b. 2.5 c. 3.0 d. 3.5

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: ______

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: ________

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: Select one: a. 2.3 b. 3.7 c. 4.1 d. 5.0

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: Select one: a. 2.0 b. 2.5 c. 3.0 d. 3.5

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? Select one:

a. The statistic proposed by Statistician A

b. The statistic proposed by Statistician B

1. ? 0

2.  

3. ? 0

4.  

5. ? 1,019

Answer 1

Estimating the Expectation

A measurement follows the normal distribution with a standard deviation of 15 and an unknown expectation μ. You can consider that measurement to be the "original" distribution. Two statisticians propose two distinct ways to estimate the unknown quantity μ with the aid of a sample of size 36. They will do that by evaluating two different SAMPLING DISTRIBUTIONS to determine which method is better. There are two statisticians involved in this task: statistician "A" and statistician "B." Statistician A proposes to use the sampling distribution of the sample average as an estimate. Statistician B proposes to use the sampling distribution of 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: The expectation of the sample mean is equal to the population mean.  The expectation of Statistician  A is 5  ?

 

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:  The estimate of standard  deviation by Statatician A is 2.5 ?

 

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.)

Select one:

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.)

Select one:

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: The expectation of the sample mean is the population mean, so x=2.5

 

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:  Standard deviation (SD) =2.3

 

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.)

Select one:

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.)

Select one:

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?

Select one:

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 handled 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

Answer Expected calls in 1 minute, E(60X) 60.E(X)=60X5.35=321.

 

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.)

Answer: 336.079

 

Question 12

The probability that the switchboard will need to handle 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.)

Answer: 0.1206 ?