Why is there a b in the equation for mean?

[inv]

[Solved]Stats question regarding mean

Homework Statement

Yo.Exam marks,X've 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've mean 55 and standard deviation 6.96.Find the values of a and b.*From the answer sheet -to form equation for mean it's 55=70a+b,why is there "b" in there?

Homework Equations

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The Attempt at a Solution

55=70a+b
U don't need this,I just need some clarity on the above * thing.Pls?

 

[HallsofIvy]

The formula for mean is "linear"- add the values and divide by n- which means that if Y= aX+ b, then mean(Y)= a*mean(X)+ b so your first equation is a(70)+ b= 55.

Formula for standard deviation is "root-mean-square":
$$\sqrt{\Sigma \frac{(X-\mu)^2}{n}}$$
so the standard deviation of Y= aX+ b is
$$\sqrt{\Sigma \frac{((aX+b)-(a\mu+b))^2}{n}}= \sqrt{\Sigma\frac{a^2(x-\mu)^2}{n}}= a\sigma$$
where $\mu$ and $\sigma$ are the mean and standard deviation of X respectively. Since the stadard deviation of X is 8.7 and you want the standard deviation of Y to be 6.96, you must have 6.96= a(8.7).

It should be easy to solve the equations 70a+ b= 55 and 8.7a= 6.96 for a and b.

 

[inv]

Thanx so much for ur stuff and kind,clear explanations.Solved.Cya.~The End~