Question 1174177: Solve for the mean μ, variance σ^2 of the population and the mean μ_¯x and variance 〖σ^2〗_¯x of the sampling distribution of the sample means.
Consider all samples of size 5 from this population: 2, 5, 6, 8, 10, 12 and 13.
Answer by CPhill(1987)   (Show Source):
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**1. Calculate the Population Mean (μ)**
* The population is: 2, 5, 6, 8, 10, 12, 13
* Sum of the population: 2 + 5 + 6 + 8 + 10 + 12 + 13 = 56
* Number of elements in the population: 7
* Population mean (μ) = Sum / Number of elements = 56 / 7 = 8
**2. Calculate the Population Variance (σ^2)**
* First, find the squared differences from the mean:
* (2 - 8)^2 = 36
* (5 - 8)^2 = 9
* (6 - 8)^2 = 4
* (8 - 8)^2 = 0
* (10 - 8)^2 = 4
* (12 - 8)^2 = 16
* (13 - 8)^2 = 25
* Sum of squared differences: 36 + 9 + 4 + 0 + 4 + 16 + 25 = 94
* Population variance (σ^2) = Sum of squared differences / Number of elements = 94 / 7 ≈ 13.4286
**3. Generate All Possible Samples of Size 5**
* We need to find all combinations of 5 elements from the population of 7.
* These samples are:
* (2, 5, 6, 8, 10)
* (2, 5, 6, 8, 12)
* (2, 5, 6, 8, 13)
* (2, 5, 6, 10, 12)
* (2, 5, 6, 10, 13)
* (2, 5, 6, 12, 13)
* (2, 5, 8, 10, 12)
* (2, 5, 8, 10, 13)
* (2, 5, 8, 12, 13)
* (2, 5, 10, 12, 13)
* (2, 6, 8, 10, 12)
* (2, 6, 8, 10, 13)
* (2, 6, 8, 12, 13)
* (2, 6, 10, 12, 13)
* (2, 8, 10, 12, 13)
* (5, 6, 8, 10, 12)
* (5, 6, 8, 10, 13)
* (5, 6, 8, 12, 13)
* (5, 6, 10, 12, 13)
* (5, 8, 10, 12, 13)
* (6, 8, 10, 12, 13)
**4. Calculate the Mean of Each Sample (Sample Means)**
* Calculate the mean of each of the samples.
* (2, 5, 6, 8, 10) mean = 6.2
* (2, 5, 6, 8, 12) mean = 6.6
* (2, 5, 6, 8, 13) mean = 6.8
* (2, 5, 6, 10, 12) mean = 7
* (2, 5, 6, 10, 13) mean = 7.2
* (2, 5, 6, 12, 13) mean = 7.6
* (2, 5, 8, 10, 12) mean = 7.4
* (2, 5, 8, 10, 13) mean = 7.6
* (2, 5, 8, 12, 13) mean = 8
* (2, 5, 10, 12, 13) mean = 8.4
* (2, 6, 8, 10, 12) mean = 7.6
* (2, 6, 8, 10, 13) mean = 7.8
* (2, 6, 8, 12, 13) mean = 8.2
* (2, 6, 10, 12, 13) mean = 8.6
* (2, 8, 10, 12, 13) mean = 9
* (5, 6, 8, 10, 12) mean = 8.2
* (5, 6, 8, 10, 13) mean = 8.4
* (5, 6, 8, 12, 13) mean = 8.8
* (5, 6, 10, 12, 13) mean = 9.2
* (5, 8, 10, 12, 13) mean = 9.6
* (6, 8, 10, 12, 13) mean = 9.8
**5. Calculate the Mean of the Sample Means (μ_¯x)**
* Sum of sample means: 6.2 + 6.6 + 6.8 + ... + 9.8 = 168
* Number of samples: 21
* Mean of sample means (μ_¯x) = 168 / 21 = 8
**6. Calculate the Variance of the Sample Means (σ^2_¯x)**
* Find the squared differences from μ_¯x (8):
* (6.2-8)^2 = 3.24, (6.6-8)^2 = 1.96, (6.8-8)^2 = 1.44, ...,(9.8-8)^2=3.24
* Sum of squared differences: 18.8
* Variance of sample means (σ^2_¯x) = 18.8 / 21 ≈ 0.8952
**Results:**
* Population Mean (μ): 8
* Population Variance (σ^2): 13.4286
* Mean of Sample Means (μ_¯x): 8
* Variance of Sample Means (σ^2_¯x): 0.8952
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