Respuesta :
To solve this problem, we need to follow the steps provided.
a. Find the value of the population standard deviation sigma.
The population consists of three values: 1, 2, and 12.
The population standard deviation (σ) can be calculated using the formula:
σ = √[Σ(x - μ)^2 / N]
where:
- x is each value in the population
- μ is the population mean
- N is the size of the population
Calculating the population mean (μ):
μ = (1 + 2 + 12) / 3 = 5
Calculating the population standard deviation (σ):
σ = √[(1 - 5)^2 + (2 - 5)^2 + (12 - 5)^2) / 3]
σ = √[(16 + 9 + 49) / 3]
σ = √(74 / 3)
σ = √24.67
σ ≈ 4.97
Therefore, the value of the population standard deviation (σ) is approximately 4.97.
b. Find the standard deviation of each of the nine samples, then summarize the sampling distribution of the standard deviations in the format of a table representing the probability distribution of the distinct standard deviation values. Use ascending order of the sample standard deviations.
The nine samples and their standard deviations are:
Sample 1: (1, 1) - Standard deviation = 0
Sample 2: (1, 2) - Standard deviation = 0.707
Sample 3: (1, 12) - Standard deviation = 7.778
Sample 4: (2, 1) - Standard deviation = 0.707
Sample 5: (2, 2) - Standard deviation = 0
Sample 6: (2, 12) - Standard deviation = 7.071
Sample 7: (12, 1) - Standard deviation = 7.778
Sample 8: (12, 2) - Standard deviation = 7.071
Sample 9: (12, 12) - Standard deviation = 0
Summarizing the sampling distribution of the standard deviations in a table:
| Standard Deviation | Probability |
|-------------------|-------------|
| 0 | 2/9 |
| 0.707 | 2/9 |
| 7.071 | 1/9 |
| 7.778 | 2/9 |
c. Find the mean of the sampling distribution of the sample standard deviations.
The mean of the sampling distribution of the sample standard deviations is calculated as:
Mean = Σ(sample standard deviations) / number of samples
Mean = (0 + 0.707 + 7.778 + 0.707 + 0 + 7.071 + 7.778 + 7.071 + 0) / 9
Mean = 31.112 / 9
Mean ≈ 3.457
Therefore, the mean of the sampling distribution of the sample standard deviations is approximately 3.457.
d. Do the sample standard deviations target the value of the population standard deviation? In general, do sample standard deviations make good estimators of population standard deviations? Why or why not?
The sample standard deviations do not directly target the value of the population standard deviation (σ ≈ 4.97). The sample standard deviations range from 0 to 7.778, with some values being close to the population standard deviation and others being quite different.
In general, sample standard deviations can be good estimators of population standard deviations, but they have some limitations:
1. Sample standard deviations are subject to sampling variability, meaning that different samples drawn from the same population can have different standard deviations.
2. As the sample size increases, the sample standard deviation becomes a better estimator of the population standard deviation.
3. The sample standard deviation is an unbiased estimator of the population standard deviation, meaning that on average, the sample standard deviation will be equal to the population standard deviation.
4. However, for small sample sizes, the sample standard deviation may not be a very accurate estimator of the population standard deviation, as it can be influenced by the specific values in the sample.
In this case, with a small sample size of 2, the sample standard deviations do
a. Find the value of the population standard deviation sigma.
The population consists of three values: 1, 2, and 12.
The population standard deviation (σ) can be calculated using the formula:
σ = √[Σ(x - μ)^2 / N]
where:
- x is each value in the population
- μ is the population mean
- N is the size of the population
Calculating the population mean (μ):
μ = (1 + 2 + 12) / 3 = 5
Calculating the population standard deviation (σ):
σ = √[(1 - 5)^2 + (2 - 5)^2 + (12 - 5)^2) / 3]
σ = √[(16 + 9 + 49) / 3]
σ = √(74 / 3)
σ = √24.67
σ ≈ 4.97
Therefore, the value of the population standard deviation (σ) is approximately 4.97.
b. Find the standard deviation of each of the nine samples, then summarize the sampling distribution of the standard deviations in the format of a table representing the probability distribution of the distinct standard deviation values. Use ascending order of the sample standard deviations.
The nine samples and their standard deviations are:
Sample 1: (1, 1) - Standard deviation = 0
Sample 2: (1, 2) - Standard deviation = 0.707
Sample 3: (1, 12) - Standard deviation = 7.778
Sample 4: (2, 1) - Standard deviation = 0.707
Sample 5: (2, 2) - Standard deviation = 0
Sample 6: (2, 12) - Standard deviation = 7.071
Sample 7: (12, 1) - Standard deviation = 7.778
Sample 8: (12, 2) - Standard deviation = 7.071
Sample 9: (12, 12) - Standard deviation = 0
Summarizing the sampling distribution of the standard deviations in a table:
| Standard Deviation | Probability |
|-------------------|-------------|
| 0 | 2/9 |
| 0.707 | 2/9 |
| 7.071 | 1/9 |
| 7.778 | 2/9 |
c. Find the mean of the sampling distribution of the sample standard deviations.
The mean of the sampling distribution of the sample standard deviations is calculated as:
Mean = Σ(sample standard deviations) / number of samples
Mean = (0 + 0.707 + 7.778 + 0.707 + 0 + 7.071 + 7.778 + 7.071 + 0) / 9
Mean = 31.112 / 9
Mean ≈ 3.457
Therefore, the mean of the sampling distribution of the sample standard deviations is approximately 3.457.
d. Do the sample standard deviations target the value of the population standard deviation? In general, do sample standard deviations make good estimators of population standard deviations? Why or why not?
The sample standard deviations do not directly target the value of the population standard deviation (σ ≈ 4.97). The sample standard deviations range from 0 to 7.778, with some values being close to the population standard deviation and others being quite different.
In general, sample standard deviations can be good estimators of population standard deviations, but they have some limitations:
1. Sample standard deviations are subject to sampling variability, meaning that different samples drawn from the same population can have different standard deviations.
2. As the sample size increases, the sample standard deviation becomes a better estimator of the population standard deviation.
3. The sample standard deviation is an unbiased estimator of the population standard deviation, meaning that on average, the sample standard deviation will be equal to the population standard deviation.
4. However, for small sample sizes, the sample standard deviation may not be a very accurate estimator of the population standard deviation, as it can be influenced by the specific values in the sample.
In this case, with a small sample size of 2, the sample standard deviations do
