Estimating the Expectation A measurement follows the normal distribution with a

Question

Estimating the Expectation A measurement follows the normal distribution with a standard deviation of 15 and an unknown expectation . You can consider that measurement to be the "original" distribution. Two statisticians propose two distinct ways to estimate the unknown quantity 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 (=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 (=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 (=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 (=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 Not yet answered Marked out of 1.00 Flag question Question text Assume that the actual expectation of the measurement is equal to 2.3 (=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 (=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 (=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 (=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: 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: ?

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: ?

Explanation / Answer

Question 1 Assume that the actual expectation of the measurement is equal to 5 (=5). Then the expectation of the statistic that was proposed by Statistician A is equal to: Answer: ?

By central limit theorem, the sample average is equal to the population mean.

So, the expectation of the statistic that was proposed by Statistician A is equal to 5.

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

By central limit theorem, the standard deviation of the mean is equal to the standard deviation divided square root of sample size.

So, the standard deviation of the statistic that was proposed by Statistician A is equal 15/sqrt(36) = 15/6 = 2.5

Question 3 Assume that the actual expectation of the measurement is equal to 5 (=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

Have taken 1000 samples of size 36, of normal distribution with mean = 5 and standard deviation = 15 and taken median of each sample. The average of the sample was 5.02241.

So, the option (d) 5.0 is correct.

Question 4 Assume that the actual expectation of the measurement is equal to 5 (=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.

Have taken 1000 samples of size 36, of normal distribution with mean = 5 and standard deviation = 15 and taken median of each sample. The average of the sample was 3.049176.

So, the option (c) 3.0 is correct.

Question 5 Not yet answered Marked out of 1.00 Flag question Question text Assume that the actual expectation of the measurement is equal to 2.3 (=2.3). Then the expectation of the statistic that was proposed by Statistician A is equal to: Answer:

By central limit theorem, the sample average is equal to the population mean.

So, the expectation of the statistic that was proposed by Statistician A is equal to 2.3.

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

By central limit theorem, the standard deviation of the mean is equal to the standard deviation divided square root of sample size.

So, the standard deviation of the statistic that was proposed by Statistician A is equal 15/sqrt(36) = 15/6 = 2.5

Question 7 Assume that the actual expectation of the measurement is equal to 2.3 (=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.

Have taken 1000 samples of size 36, of normal distribution with mean = 2.3 and standard deviation = 15 and taken median of each sample. The average of the sample was 2.335652.

So, the option (a) 2.3 is correct.

Question 8 Assume that the actual expectation of the measurement is equal to 2.3 (=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.

Have taken 1000 samples of size 36, of normal distribution with mean = 2.3 and standard deviation = 15 and taken median of each sample. The average of the sample was 3.048367.

So, the option (c) 3.0 is correct.

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

In order to choose between the two options they agree to prefer the statistic that has a smaller variance or standard deviation (with respect to the sampling distribution).

The standard deviation of the statistic that was proposed by Statistician A is 2.5 and the standard deviation of the statistic that was proposed by Statistician B is 3.0. As, the standard deviation of the statistic that was proposed by Statistician A is less than standard deviation of the statistic that was proposed by Statistician B, option A is correct.

a. The statistic proposed by Statistician A

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