Can someone please help with this?
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
To answer these questions, we need to understand the concepts of sample mean and sample median, as well as the properties of their sampling distributions.
1. The expectation of the sample mean (statistic proposed by Statistician A) can be found using the formula: E(X̄) = μ, where X̄ is the sample mean and μ is the population mean. Therefore, the expectation of the statistic proposed by Statistician A is equal to the population mean, which is 5.
2. The standard deviation of the sample mean (statistic proposed by Statistician A) can be found using the formula: σ(X̄) = σ/√n, where σ is the standard deviation of the original distribution and n is the sample size. Plugging in the values, we have σ(X̄) = 15/√36 = 5/2 = 2.5. Therefore, the standard deviation of the statistic proposed by Statistician A is 2.5.
3. To find the expectation of the sample median (statistic proposed by Statistician B), we need to consider the standard normal distribution. The expectation of the sample median can be approximated as μ + 1.253σ, where μ is the population mean and σ is the standard deviation. Plugging in the values, we have 5 + 1.253(15) ≈ 5 + 18.8 ≈ 23.8. Therefore, the expectation of the statistic proposed by Statistician B is approximately 23.8.
4. To find the standard deviation of the sample median (statistic proposed by Statistician B), we use a similar formula but with a slightly different constant: σ(M) ≈ σ/√(n/2), where M is the sample median. Plugging in the values, we have σ(M) ≈ 15/√(36/2) ≈ 15/√18 ≈ 15/4.24 ≈ 3.53. Therefore, the standard deviation of the statistic proposed by Statistician B is approximately 3.53.
5. The expectation of the sample mean (statistic proposed by Statistician A) is the population mean, so it remains unchanged at 2.3.
6. The standard deviation of the sample mean (statistic proposed by Statistician A) is still given by σ(X̄) = σ/√n. Plugging in the values, we have σ(X̄) = 15/√36 = 5/2 = 2.5. Therefore, the standard deviation of the statistic proposed by Statistician A remains 2.5.
7. The expectation of the sample median (statistic proposed by Statistician B) remains unchanged at 2.3.
8. The standard deviation of the sample median (statistic proposed by Statistician B) remains the same as before, approximately 3.53.
9. Comparing the standard deviations of the two statistics, we can see that the statistic proposed by Statistician A (sample mean) has a smaller standard deviation of 2.5, while the statistic proposed by Statistician B (sample median) has a larger standard deviation of approximately 3.53. A smaller standard deviation implies that the values tend to be more concentrated around the expectation of the measurement. Therefore, the statistic proposed by Statistician A produces values that tend to be more concentrated about the expectation of the measurement.
In conclusion, the correct answer to Task 9 is:
a. The statistic proposed by Statistician A
1. The expectation of the statistic proposed by Statistician A is equal to the actual expectation of the measurement, which is μ=5.
2. The standard deviation of the statistic proposed by Statistician A is equal to the standard deviation of the original distribution divided by the square root of the sample size: 15 / √36 = 15 / 6 = 2.5.
3. The expectation of the statistic proposed by Statistician B is equal to the actual expectation of the measurement, which is μ=5.
4. The standard deviation of the statistic proposed by Statistician B is equal to the standard deviation of the original distribution divided by a constant factor, which depends on the sample size and the shape of the original distribution. Without additional information, it is not possible to determine the exact standard deviation.
5. The expectation of the statistic proposed by Statistician A is equal to the actual expectation of the measurement, which is μ=2.3.
6. The standard deviation of the statistic proposed by Statistician A is equal to the standard deviation of the original distribution divided by the square root of the sample size: 15 / √36 = 15 / 6 = 2.5.
7. The expectation of the statistic proposed by Statistician B is equal to the actual expectation of the measurement, which is μ=2.3.
8. The standard deviation of the statistic proposed by Statistician B is equal to the standard deviation of the original distribution divided by a constant factor, which depends on the sample size and the shape of the original distribution. Without additional information, it is not possible to determine the exact standard deviation.
9. Based on the information provided, both statistics have the same expectation as the actual expectation of the measurement. However, the statistic proposed by Statistician A has a smaller standard deviation (2.5) compared to the statistic proposed by Statistician B. Therefore, the statistic proposed by Statistician A produces values that tend to be more concentrated about the expectation of the measurement. Thus, the answer is a. The statistic proposed by Statistician A.