Question 1174175: TASK 1: 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.
1. A population consists of the five measurements: 2, 6, 8, 0 and 3. Suppose samples of size 2 are drawn from this population.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's solve this problem step-by-step.
**1. Calculate the Population Mean (μ)**
* Population: 2, 6, 8, 0, 3
* Sum of the population: 2 + 6 + 8 + 0 + 3 = 19
* Number of elements in the population: 5
* Population mean (μ) = Sum / Number of elements = 19 / 5 = 3.8
**2. Calculate the Population Variance (σ^2)**
* First, find the squared differences from the mean:
* (2 - 3.8)^2 = (-1.8)^2 = 3.24
* (6 - 3.8)^2 = (2.2)^2 = 4.84
* (8 - 3.8)^2 = (4.2)^2 = 17.64
* (0 - 3.8)^2 = (-3.8)^2 = 14.44
* (3 - 3.8)^2 = (-0.8)^2 = 0.64
* Sum of squared differences: 3.24 + 4.84 + 17.64 + 14.44 + 0.64 = 40.8
* Population variance (σ^2) = Sum of squared differences / Number of elements = 40.8 / 5 = 8.16
**3. Generate All Possible Samples of Size 2**
* We need to find all combinations of 2 elements from the population of 5.
* These samples are:
* (2, 6)
* (2, 8)
* (2, 0)
* (2, 3)
* (6, 8)
* (6, 0)
* (6, 3)
* (8, 0)
* (8, 3)
* (0, 3)
**4. Calculate the Mean of Each Sample (Sample Means)**
* Calculate the mean of each of the samples.
* (2, 6) mean = 4
* (2, 8) mean = 5
* (2, 0) mean = 1
* (2, 3) mean = 2.5
* (6, 8) mean = 7
* (6, 0) mean = 3
* (6, 3) mean = 4.5
* (8, 0) mean = 4
* (8, 3) mean = 5.5
* (0, 3) mean = 1.5
**5. Calculate the Mean of the Sample Means (μ_¯x)**
* Sum of sample means: 4 + 5 + 1 + 2.5 + 7 + 3 + 4.5 + 4 + 5.5 + 1.5 = 38
* Number of samples: 10
* Mean of sample means (μ_¯x) = 38 / 10 = 3.8
**6. Calculate the Variance of the Sample Means (σ^2_¯x)**
* Find the squared differences from μ_¯x (3.8):
* (4 - 3.8)^2 = 0.04
* (5 - 3.8)^2 = 1.44
* (1 - 3.8)^2 = 7.84
* (2.5 - 3.8)^2 = 1.69
* (7 - 3.8)^2 = 10.24
* (3 - 3.8)^2 = 0.64
* (4.5 - 3.8)^2 = 0.49
* (4 - 3.8)^2 = 0.04
* (5.5 - 3.8)^2 = 2.89
* (1.5 - 3.8)^2 = 5.29
* Sum of squared differences: 0.04 + 1.44 + 7.84 + 1.69 + 10.24 + 0.64 + 0.49 + 0.04 + 2.89 + 5.29 = 30.6
* Variance of sample means (σ^2_¯x) = 30.6 / 10 = 3.06
**Results:**
* Population Mean (μ): 3.8
* Population Variance (σ^2): 8.16
* Mean of Sample Means (μ_¯x): 3.8
* Variance of Sample Means (σ^2_¯x): 3.06
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