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Two data sets each have 5 values. Their variances are 4 and 5, and their means are 2 and 4 respectively. Find twice the variance of the combined data set.

1. 13
2. 12
3. 11
4. 10

Correct answer: 13

Solution

n1 = n2 = 5, mean1 = 2, mean2 = 4, var1 = 4, var2 = 5. Combined mean = (5*2 + 5*4)/(5+5) = 30/10 = 3. Sum of squares for set 1: sum(xi²) = n1*(var1 + mean1²) = 5*(4+4) = 40. Sum of squares for set 2: sum(yj²) = n2*(var2 + mean2²) = 5*(5+16) = 105. Combined sum of squares = 40 + 105 = 145. Combined variance = 145/10 - 3² = 14.5 - 9 = 5.5. Twice the variance = 2 * 5.5 = 11. But options suggest 13. Let me recheck: combined variance formula: sigma² = [n1*(sigma1² + d1²) + n2*(sigma2² + d2²)] / (n1+n2), where d1 = mean1 - combined_mean = 2-3 = -1, d2 = mean2 - combined_mean = 4-3 = 1. = [5*(4+1) + 5*(5+1)] / 10 = [5*5 + 5*6]/10 = [25+30]/10 = 55/10 = 5.5. 2 * 5.5 = 11. Answer should be 11.

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