Question

If the standard deviation of six numbers \(x_1,x_2,x_3,x_4,x_5,x_6\) is \(4\), then the variance of \(2x_1+3,2x_2+3,2x_3+3,2x_4+3,2x_5+3,2x_6+3\) is

• A. \(64\)
• B. \(67\)
• C. \(16\)
• D. \(19\)
• E. \(8\)

Hint:

If every observation is multiplied by \(a\), the variance gets multiplied by \(a^2\). Adding or subtracting a constant does not change the variance.

Answer 1

The Correct Option is A

Step 1: Recall the relation between standard deviation and variance.
We know that:
\[ \text{Variance}=(\text{Standard Deviation})^2 \] Since the standard deviation of the original six numbers is \(4\), their variance is:
\[ 4^2=16 \]

Step 2: Write the transformation of the variables.
Each new number is of the form:
\[ y_i=2x_i+3 \] So the new set is obtained by multiplying each original observation by \(2\) and then adding \(3\).

Step 3: Recall how variance changes under linear transformation.
If \[ y=ax+b, \] then the variance changes as:
\[ \operatorname{Var}(y)=a^2\operatorname{Var}(x) \] The constant term \(b\) does not affect the variance.

Step 4: Identify the values of \(a\) and \(b\).
Here, \[ a=2 \quad \text{and} \quad b=3 \] Therefore:
\[ \operatorname{Var}(2x+3)=2^2\operatorname{Var}(x) \]

Step 5: Substitute the original variance.
Since the original variance is \(16\), we get:
\[ \operatorname{Var}(2x+3)=4\times 16 \]

Step 6: Simplify.
\[ 4\times 16=64 \]

Step 7: State the final answer.
Hence, the variance of \(2x_1+3,2x_2+3,2x_3+3,2x_4+3,2x_5+3,2x_6+3\) is:
\[ \boxed{64} \] which matches option \((1)\).