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The standard deviation of 20 observations x1, x2,..., x20 is 4 and that of another 20 observations y1, y2,..., y20 is 3. Define X_i = (x_i - x_bar)*(y_i - y_bar) where x_bar and y_bar are the respective means. If the sum of X_i from i=1 to 20 equals 90, then the standard deviation of the 20 observations (x1-y1, x2-y2,..., x20-y20) is divisible by

1. 5
2. 4
3. 3
4. 7

Correct answer: 5

Solution

Variance of (x_i - y_i) = Var(x) + Var(y) - 2*Cov(x,y). Var(x)=16, Var(y)=9. Cov(x,y) = (1/20)*sum(X_i) = 90/20 = 4.5. Var(x-y) = 16 + 9 - 9 = 16. SD = 4. Among the options, 4 divides 4 and also... wait: SD=4 is divisible by 4 (option B) and by 1. But also we must check if other options divide 4: 5 does not divide 4. The question asks which option the SD is divisible by. SD=4 is divisible by 4. But wait: recompute Cov: population cov = (1/n)*sum((x_i - x_bar)(y_i-y_bar)) = 90/20 = 4.5. Var(x-y) = 16+9-2*4.5 = 25-9=16. SD=4. Divisible by 4. Answer: 4.

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