Question

Let $x_1, x_2, x_3 \dots x_k$ be $k$ observations and $w_i = ax_i + b$ for $i = 1,2,3 \dots k$, where $a$ and $b$ are constants. If mean of $x_i$ is 52 and their standard deviation is 12 and mean of $w_i$ is 60 and their standard deviation is 15, then the value of $a$ and $b$ should be

Answer 1

a = 5/4, b = -5

Answer 2

Given:

$$\bar{x} = 52, \quad \sigma_x = 12,\quad \bar{w} = 60, \quad \sigma_w = 15, \quad w = ax + b.$$

1. Mean Transformation:

   $$a(52) + b = 60 \quad \Rightarrow \quad 52a + b = 60. \quad (1)$$

2. Standard Deviation Transformation:

   $$\sigma_w = |a|\sigma_x \quad \Rightarrow \quad 15 = |a|\cdot 12 \quad \Rightarrow \quad |a| = \frac{15}{12} = \frac{5}{4}.$$

   Assuming a positive scaling factor (as typically taken in such cases), we have:

   $$a = \frac{5}{4}.$$

3. Find $b$: Substitute $a = \frac{5}{4}$ into equation (1):

   $$52\left(\frac{5}{4}\right) + b = 60 \quad \Rightarrow \quad 65 + b = 60 \quad \Rightarrow \quad b = 60 - 65 = -5.$$