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

 

[PeroK]

Why a = -0.8 is rejected?

Why do you think?

 

[songoku]

Why do you think?

In my opinion, there are two possible pairs of a and b so a = -0.8 is possible

 

[PeroK]

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?

 

[FactChecker]

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.

 

[Mark44]

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.

 

[PeroK]

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]

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

 

[PeroK]

And the first one now will later be last ...