A statistics instructor compiles the exam scores of 50 students in Dataset S. The dataset … — GRE General Test

A statistics instructor compiles the exam scores of 50 students in Dataset $S$ . The dataset has a range $R$ , an interquartile range $IQR$ , and a standard deviation $s$ . A revised dataset $T$ is created by multiplying each score in Dataset $S$ by $-1.5$ and then adding $25$ to each result. Which of the following expressions correctly states the measures of dispersion for Dataset $T$ in terms of the measures of dispersion for Dataset $S$ ?

A
$\text{Range}_T = -1.5R + 25$ , $IQR_T = -1.5 IQR + 25$ , and $s_T = -1.5s + 25$
B
$\text{Range}_T = 1.5R + 25$ , $IQR_T = 1.5 IQR + 25$ , and $s_T = 1.5s + 25$
$\text{Range}_T = 1.5R$ , $IQR_T = 1.5 IQR$ , and $s_T = 1.5s$ Cevap
D
$\text{Range}_T = 1.5R$ , $IQR_T = 1.5 IQR$ , and $s_T = 2.25s$
E
$\text{Range}_T = 1.5R$ , $IQR_T = IQR$ , and $s_T = 1.5s$

Cevap

The measures of dispersion for Dataset T are given by Range_T = 1.5R, IQR_T = 1.5 IQR, and s_T = 1.5s.

When a linear transformation

y = ax + b

is applied to a dataset, any measure of dispersion

D

(such as range, interquartile range, or standard deviation) transforms according to

D_y = |a| D_x

. The additive constant

b = 25

shifts all values equally and does not change the distances between data points, so it has no effect on dispersion. The multiplicative factor

a = -1.5

scales all distances by

|-1.5| = 1.5

. Therefore,

\text{Range}_T = 1.5R

IQR_T = 1.5 IQR

, and

s_T = 1.5s

Adım Adım Çözüm

Analyze the general effect of a linear transformation y = ax + b on measures of dispersion.

Measures of dispersion (range, interquartile range, standard deviation) quantify spread and distance between data points. Adding a constant b shifts the whole distribution without altering distance between points, so b has zero impact on spread.

Additive constants shift position, not dispersion.

Determine the effect of multiplying by scale factor a = -1.5 on range, IQR, and standard deviation.

Multiplying data by a constant scale factor a scales all distance-based measures by |a|. Since dispersion measures must be non-negative, the scale factor used is |-1.5| = 1.5.

Distances between values scale by the absolute value of the multiplicative factor.

Combine the results to state Range_T, IQR_T, and s_T in terms of R, IQR, and s.

Range_T = 1.5R, IQR_T = 1.5 IQR, and s_T = 1.5s.

All three metrics scale by 1.5 and are unaffected by the +25 term.

Anahtar Kavram

Linear Transformations of Dispersion Metrics

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