Question 1184283: Two samples, sizes 40 and 70 respectively, are taken from a population with unknown
mean μ and unknown variance σ^2 . The data is shown below.
Sample I
x1 1 2 3 4 5
f1 2 5 18 12 3
Sample 2
x2 1 2 3 4 5 6 7
f2 3 6 12 26 17 5 1
Using the data from the two samples above, obtain unbiased estimates of
i. The population mean μ
ii. The population variance σ^2
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to calculate the unbiased estimates for the population mean (μ) and variance (σ²) using the given data from two samples:
**i. Estimating the Population Mean (μ)**
Since we have two samples, we'll calculate the weighted average of the sample means to get the best unbiased estimate of the population mean.
* **Sample 1:**
- Sum of (x1 * f1) = (1*2) + (2*5) + (3*18) + (4*12) + (5*3) = 2 + 10 + 54 + 48 + 15 = 129
- n1 (sample size) = 2 + 5 + 18 + 12 + 3 = 40
- Sample mean (x̄1) = (Sum of (x1 * f1)) / n1 = 129 / 40 = 3.225
* **Sample 2:**
- Sum of (x2 * f2) = (1*3) + (2*6) + (3*12) + (4*26) + (5*17) + (6*5) + (7*1) = 3 + 12 + 36 + 104 + 85 + 30 + 7 = 277
- n2 (sample size) = 3 + 6 + 12 + 26 + 17 + 5 + 1 = 70
- Sample mean (x̄2) = (Sum of (x2 * f2)) / n2 = 277 / 70 = 3.957 (approximately)
* **Combined Estimate of μ:**
- μ̂ = (n1 * x̄1 + n2 * x̄2) / (n1 + n2) = (40 * 3.225 + 70 * 3.957) / (40 + 70) = (129 + 277) / 110 = 406 / 110 = 3.691 (approximately)
**ii. Estimating the Population Variance (σ²)**
We'll use the sample variances and combine them in a weighted average, but first, we need to calculate the sample variances.
* **Sample 1:**
- Calculate the sum of (f1 * (x1 - x̄1)²) = 2*(1-3.225)² + 5*(2-3.225)² + 18*(3-3.225)² + 12*(4-3.225)² + 3*(5-3.225)² = 10.126 + 7.563 + 0.882 + 6.008 + 9.677 = 34.256
- Sample variance (s1²) = (Sum of (f1 * (x1 - x̄1)²)) / (n1 - 1) = 34.256 / 39 = 0.878 (approximately)
* **Sample 2:**
- Calculate the sum of (f2 * (x2 - x̄2)²) = 3*(1-3.957)² + 6*(2-3.957)² + 12*(3-3.957)² + 26*(4-3.957)² + 17*(5-3.957)² + 5*(6-3.957)² + 1*(7-3.957)² = 26.65 + 11.45 + 10.92 + 0.07 + 19.34 + 21.08 + 9.27 = 98.78
- Sample variance (s2²) = (Sum of (f2 * (x2 - x̄2)²)) / (n2 - 1) = 98.78 / 69 = 1.432 (approximately)
* **Combined Estimate of σ²:**
- σ̂² = [(n1 - 1) * s1² + (n2 - 1) * s2²] / (n1 + n2 - 2) = (39 * 0.878 + 69 * 1.432) / (40 + 70 - 2) = (34.242 + 98.808) / 108 = 133.05 / 108 = 1.232 (approximately)
**Therefore:**
* **i. Unbiased estimate of μ:** 3.691 (approximately)
* **ii. Unbiased estimate of σ²:** 1.232 (approximately)
|
|
|
