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Given that sum(i=1 to 5) of (x_i - 10) = 5 and sum(i=1 to 5) of (x_i - 10)² = 25, find the standard deviation of the data set: 2*x₁ + 7, 2*x₂ + 7, 2*x₃ + 7, 2*x₄ + 7, 2*x₅ + 7.
- 8
- 16
- 4
- 2
Correct answer: 4
Solution
Let y_i = x_i - 10. Sum(y_i) = 5, so mean of y = 1. Sum(y_i²) = 25. Variance of y = mean(y²) - (mean y)² = 25/5 - 1² = 5 - 1 = 4. SD(y) = SD(x) = 2 (since y = x - 10, a shift). For the new dataset 2x_i + 7: SD(2x+7) = |2|*SD(x) = 2*2 = 4.
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