# Finding constant related to random variable

songoku
Homework Statement:
Exam marks, X, have mean 70 and standard deviation 8.7. The marks need to be scaled using the formula Y = aX + b so that the scaled marks, Y, have mean 55 and standard deviation 6.96. Find the values of a and b
Relevant Equations:
E(aX + b) = a.E(X) + b

Var(aX + b) = a^2 Var (X)
Var (Y) = a2 . Var (X)
(6.96)2 = a2 . (8.7)2
a = ± 0.8

But the answer key states that the value of a is only 0.8

Why a = -0.8 is rejected? Thanks

Homework Helper
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Why a = -0.8 is rejected?
Why do you think?

songoku
songoku
Why do you think?
In my opinion, there are two possible pairs of a and b so a = -0.8 is possible

Homework Helper
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In my opinion, there are two possible pairs of a and b so a = -0.8 is possible
What do you want the mean to be?

songoku
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You are correct in saying that the -0.8 would give the correct mean and variance, but the term "scaled" is assumed to imply that the values of X are just shifted and stretched/shrunk -- not reversed.

songoku, PeroK and Delta2
Mentor
Why a = -0.8 is rejected?
I would be very angry if my raw score (X) on the exam was 70, but my adjusted score wound up being -56.

LastScattered1090, FactChecker and songoku
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I would be very angry if my raw score (X) on the exam was 70, but my adjusted score wound up being -56.
The proposed formula is ##Y = -0.8X + 111##. Your score of##70## would, indeed, scale to the average of ##55##. But a score of ##80## would scale to ##47##.

songoku
songoku
What do you want the mean to be?
For a = -0.8 and b = 111, the mean for Y is 55

I would be very angry if my raw score (X) on the exam was 70, but my adjusted score wound up being -56.
Well, I won't because my peers' scores will also be negative

Besides, there is still value of b to make the scaled mark positive

You are correct in saying that the -0.8 would give the correct mean and variance, but the term "scaled" is assumed to imply that the values of X are just shifted and stretched/shrunk -- not reversed.
I understand the assumption

Edit: I just saw post #7. That is indeed does not make sense

Thank you for all the help and explanation PeroK, FactChecker, Mark44