Let 1`x_(1),x_(2)….x_(n)` be n obervations .Let `w_(i)=lx_(i) +k " for " i=1,2….n,`where l and k are constants. If the mean of `x_(i)` is 48 and their standard deviation is 12 the mean of `w_(i)` 's is 55 and standard deviation of `w_(i) ` is 15 then the value of l and k should be

Options

A. l=1.25,k=-5
B. l=-1.25,k=5
C. l=2.5,k= -5
D. l=2.5, k=5

Explanation

To solve the problem, we need to find the values of constants \( l \) and \( k \) given the transformations of the observations \( x_i \) into \( w_i \) using the equation \( w_i = l x_i + k \).

### Step-by-step Solution:

1. **Understanding the Mean Transformation**:
The mean of \( w_i \) can be expressed in terms of the mean of \( x_i \):
\[ \bar{w} = l \bar{x} + k \]
Given:
- Mean of \( x_i \) (\( \bar{x} \)) = 48
- Mean of \( w_i \) (\( \bar{w} \)) = 55

Substituting these values into the equation gives:
\[ 55 = l \cdot 48 + k \quad \text{(Equation 1)} \]

2. **Understanding the Standard Deviation Transformation**:
The standard deviation of \( w_i \) can be expressed in terms of the standard deviation of \( x_i \):
\[ s_w = |l| s_x \]
Given:
- Standard deviation of \( x_i \) (\( s_x \)) = 12
- Standard deviation of \( w_i \) (\( s_w \)) = 15

Substituting these values into the equation gives:
\[ 15 = |l| \cdot 12 \]

3. **Solving for \( l \)**:
From the standard deviation equation:
\[ |l| = \frac{15}{12} = 1.25 \]
Since \( l \) is a constant, we can take \( l = 1.25 \).

4. **Substituting \( l \) back into Equation 1**:
Now we substitute \( l = 1.25 \) back into Equation 1 to find \( k \):
\[ 55 = 1.25 \cdot 48 + k \]
Calculating \( 1.25 \cdot 48 \):
\[ 1.25 \cdot 48 = 60 \]
Thus, we have:
\[ 55 = 60 + k \]

5. **Solving for \( k \)**:
Rearranging gives:
\[ k = 55 - 60 = -5 \]

### Final Values:
- \( l = 1.25 \)
- \( k = -5 \)