Moderate -0.8 This is a straightforward application of linear transformation properties for mean and standard deviation. Students need only recall that E(aX+b) = aE(X)+b and SD(aX+b) = |a|SD(X), then solve two simple equations. It requires basic algebraic manipulation with no conceptual difficulty or problem-solving insight.
1 Exam marks, \(X\), have mean 70 and standard deviation 8.7. The marks need to be scaled using the formula \(Y = a X + b\) so that the scaled marks, \(Y\), have mean 55 and standard deviation 6.96. Find the values of \(a\) and \(b\).
For an equation relating to the variance or sd, only \(a\) in it
\(a = 0.8\)
A1
For correct \(a\)
\(b = -1\)
A1 (4)
For correct \(b\)
$55 = 70a + b$ | M1 | For an equation relating to the means
$6.96 = 8.7a$ or $6.96^2 = 8.7^2 a^2$ | M1 | For an equation relating to the variance or sd, only $a$ in it
$a = 0.8$ | A1 | For correct $a$
$b = -1$ | A1 (4) | For correct $b$
1 Exam marks, $X$, have mean 70 and standard deviation 8.7. The marks need to be scaled using the formula $Y = a X + b$ so that the scaled marks, $Y$, have mean 55 and standard deviation 6.96. Find the values of $a$ and $b$.
\hfill \mbox{\textit{CAIE S2 2005 Q1 [4]}}