Data Set A consists of n numbers with mean \mu and standard deviation \sigma, where \sigma… — GMAT

Data Set $A$ consists of $n$ numbers with mean $\mu$ and standard deviation $\sigma$ , where $\sigma > 0$ . Data Set $B$ is created by transforming each element $x$ in Data Set $A$ into $y = 5 - 2x$ . If the variance of Data Set $B$ is equal to $k$ , which of the following expressions represents the standard deviation of Data Set $A$ in terms of $k$ ?

$\frac{\sqrt{k}}{2}$ Cevap
B
$-\frac{\sqrt{k}}{2}$
C
$\frac{k}{4}$
D
$\frac{\sqrt{k - 5}}{2}$
E
$\sqrt{\frac{k}{2}}$

Cevap

\frac{\sqrt{k}}{2}

Under a linear transformation

y = ax + b

, constant shifts do not change data dispersion, while multiplying elements by

a

scales the standard deviation by

|a|

and the variance by

a^2

. Here,

y = -2x + 5

, so

a = -2

. The variance of Data Set

B

\text{Var}(B) = (-2)^2 \cdot \text{Var}(A) = 4\sigma^2

. Given

\text{Var}(B) = k

, we get

4\sigma^2 = k

, which leads to

\sigma^2 = \frac{k}{4}

. Taking the positive square root yields

\sigma = \frac{\sqrt{k}}{2}

Adım Adım Çözüm

Analyze the linear transformation rule for standard deviation and variance.

For

y = ax + b

\text{SD}(y) = |a| \cdot \text{SD}(x)

and

\text{Var}(y) = a^2 \cdot \text{Var}(x)

Adding a constant

b

shifts all data points equally without altering dispersion, while multiplying by

a

scales distance from the mean by

|a|

Apply the transformation

y = -2x + 5

to find the variance of Data Set

B

in terms of

\sigma

\text{Var}(B) = (-2)^2 \cdot \text{Var}(A) = 4\sigma^2.

The coefficient of

x

a = -2

, so variance scales by

a^2 = (-2)^2 = 4

Set the variance equal to

k

and solve for the standard deviation

\sigma

of Data Set

A

4\sigma^2 = k \implies \sigma^2 = \frac{k}{4} \implies \sigma = \frac{\sqrt{k}}{2}.

Standard deviation is strictly non-negative (

\sigma > 0

), requiring the positive square root.

Anahtar Kavram

Linear Transformation of Dispersion Measures

Tahmini Süre:2m 0s

Soru

Cevap

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