# Why do we subtract the variance?

This is not a question for doing my homework. This is a question to understand the deeper meaning of the answer. So in part b), it subtracts the variance. Why do we subtract variance and what does it mean to subtract variance? I understood variance as the distance the numbers are spread apart, so what does subtracting that mean?

Question:

Suppose that 30% of all students who have to buy a text for a particular course want a new copy (the successes!), whereas the other 70% want a used copy. Consider randomly selecting 25 purchasers. a. What are the mean value and standard deviation of the number who want a new copy of the book? b. What is the probability that the number who want new copies is more than two standard deviations away from the mean value?

X ~ Bin(25,.3)

a. E(X) = np = 7.5; Var(X) = npq = 5.25 → SD(X) = 2.29

b. P(|X – 5.25| > 2(2.29)) = P(X < 0.67 or X > 9.83) = P(X = 0) + P(X > 9.83) = b(0;25,.3) + 1 – P(X ≤ 9) = b(0;25,.3) + 1 – B(9;25,.3) = .000 + 1 – .811 = .189

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The correct answer is $$P(|X – 7.5| > 2(2.29)) = P(X < 2.92\mbox{ or }X > 12.08)=.02643 .$$
Aren't you still subtracting the variance? Since variance is 5.25. You meant subtract 7.5 right? – Doug Apr 15 '11 at 23:56