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Question

# Remember back in Chapter 5 when we used the equation$SAT = 40 \times ACT + 150$to convert an ACT score into a SAT score. Let's use this transformation again, now with random variables. Suppose your school has a mean ACT score of 29. What would its equivalent mean SAT score be?

Solution

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Given:

$SAT=40\times ACT+150$

$E(ACT)=29$

Properties mean and variance:

$E(aX+b)=aE(X)+b$

$Var(aX+b)=a^2Var(X)$

$E(X\pm Y)=E(X)\pm E(Y)$

$Var(X\pm Y)=Var(X)+Var(Y)$

Using the properties for the mean and variance, we then obtain:

$E(SAT)=E(40\times ACT+150)=40E(ACT)+150=40(29)+150=1310$

Thus the equivalent mean SAT score is 1310.

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