document.write( "Question 1184283: Two samples, sizes 40 and 70 respectively, are taken from a population with unknown
\n" ); document.write( "mean μ and unknown variance σ^2 . The data is shown below.
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\n" ); document.write( "Using the data from the two samples above, obtain unbiased estimates of
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Algebra.Com's Answer #849861 by CPhill(1959)\"\" \"About 
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Here's how to calculate the unbiased estimates for the population mean (μ) and variance (σ²) using the given data from two samples:\r
\n" ); document.write( "\n" ); document.write( "**i. Estimating the Population Mean (μ)**\r
\n" ); document.write( "\n" ); document.write( "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.\r
\n" ); document.write( "\n" ); document.write( "* **Sample 1:**
\n" ); document.write( " - Sum of (x1 * f1) = (1*2) + (2*5) + (3*18) + (4*12) + (5*3) = 2 + 10 + 54 + 48 + 15 = 129
\n" ); document.write( " - n1 (sample size) = 2 + 5 + 18 + 12 + 3 = 40
\n" ); document.write( " - Sample mean (x̄1) = (Sum of (x1 * f1)) / n1 = 129 / 40 = 3.225\r
\n" ); document.write( "\n" ); document.write( "* **Sample 2:**
\n" ); document.write( " - 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
\n" ); document.write( " - n2 (sample size) = 3 + 6 + 12 + 26 + 17 + 5 + 1 = 70
\n" ); document.write( " - Sample mean (x̄2) = (Sum of (x2 * f2)) / n2 = 277 / 70 = 3.957 (approximately)\r
\n" ); document.write( "\n" ); document.write( "* **Combined Estimate of μ:**
\n" ); document.write( " - μ̂ = (n1 * x̄1 + n2 * x̄2) / (n1 + n2) = (40 * 3.225 + 70 * 3.957) / (40 + 70) = (129 + 277) / 110 = 406 / 110 = 3.691 (approximately)\r
\n" ); document.write( "\n" ); document.write( "**ii. Estimating the Population Variance (σ²)**\r
\n" ); document.write( "\n" ); document.write( "We'll use the sample variances and combine them in a weighted average, but first, we need to calculate the sample variances.\r
\n" ); document.write( "\n" ); document.write( "* **Sample 1:**
\n" ); document.write( " - 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
\n" ); document.write( " - Sample variance (s1²) = (Sum of (f1 * (x1 - x̄1)²)) / (n1 - 1) = 34.256 / 39 = 0.878 (approximately)\r
\n" ); document.write( "\n" ); document.write( "* **Sample 2:**
\n" ); document.write( " - 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
\n" ); document.write( " - Sample variance (s2²) = (Sum of (f2 * (x2 - x̄2)²)) / (n2 - 1) = 98.78 / 69 = 1.432 (approximately)\r
\n" ); document.write( "\n" ); document.write( "* **Combined Estimate of σ²:**
\n" ); document.write( " - 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)\r
\n" ); document.write( "\n" ); document.write( "**Therefore:**\r
\n" ); document.write( "\n" ); document.write( "* **i. Unbiased estimate of μ:** 3.691 (approximately)
\n" ); document.write( "* **ii. Unbiased estimate of σ²:** 1.232 (approximately)
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