Mean and standard deviations of 10 observations are 20 and 2 respectively. If p `(p ne 0)` is multiplied to each observation and then q `(q ne 0)` is subtracted then new mean and standard deviation becomes half of original value . Then find q

Options

A. 10
B. `-10`
C. `-20`
D. `5`

Answer

To solve the problem step by step, we will follow the given conditions and derive the necessary equations.

### Step 1: Understand the given data
We have:
- Mean (x̄) = 20
- Standard deviation (σ) = 2
- Number of observations (n) = 10

### Step 2: Set up the equations for the new mean
When we multiply each observation by \( p \) and subtract \( q \), the new mean \( \bar{y} \) can be expressed as:
\[ \bar{y} = p \cdot \bar{x} - q \]
Given that the new mean is half of the original mean:
\[ \bar{y} = \frac{1}{2} \cdot 20 = 10 \]
Substituting the values into the equation:
\[ 10 = p \cdot 20 - q \]
This simplifies to:
\[ 10 = 20p - q \quad \text{(Equation 1)} \]

### Step 3: Set up the equations for the new standard deviation
The new standard deviation \( \sigma' \) can be expressed as:
\[ \sigma' = |p| \cdot \sigma \]
Given that the new standard deviation is half of the original standard deviation:
\[ \sigma' = \frac{1}{2} \cdot 2 = 1 \]
Substituting the value of σ:
\[ 1 = |p| \cdot 2 \]
This simplifies to:
\[ |p| = \frac{1}{2} \]
Thus, \( p \) can be \( \frac{1}{2} \) or \( -\frac{1}{2} \).

### Step 4: Solve for \( q \) using the values of \( p \)
#### Case 1: If \( p = \frac{1}{2} \)
Substituting \( p \) into Equation 1:
\[ 10 = 20 \cdot \frac{1}{2} - q \]
This simplifies to:
\[ 10 = 10 - q \]
Thus:
\[ q = 0 \]
Since \( q \neq 0 \), this case is not acceptable.

#### Case 2: If \( p = -\frac{1}{2} \)
Substituting \( p \) into Equation 1:
\[ 10 = 20 \cdot \left(-\frac{1}{2}\right) - q \]
This simplifies to:
\[ 10 = -10 - q \]
Thus:
\[ q = -20 \]

### Conclusion
The value of \( q \) is \( -20 \).