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Given sum_(i=1)⁵ (x_i - 10) = 5 and sum_(i=1)⁵ (x_i - 10)² = 25, find the standard deviation of the five observations 2x₁ + 7, 2x₂ + 7, 2x₃ + 7, 2x₄ + 7, 2x₅ + 7.
- 8
- 16
- 4
- 2
Correct answer: 4
Solution
Let a_i = x_i - 10. Given: sum a_i = 5, sum a_i² = 25. Mean of a_i = 5/5 = 1. Variance of a_i = (1/5)*sum a_i² - (mean)² = 25/5 - 1 = 5 - 1 = 4. SD of x_i = sqrt(4) = 2 (since shifting by 10 doesn't change SD). For y_i = 2x_i + 7: SD(y_i) = |2| * SD(x_i) = 2 * 2 = 4.
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