A researcher recorded the annual rainfall totals (in inches) for a specific region over a … — GRE General Test

A researcher recorded the annual rainfall totals (in inches) for a specific region over a 20-year period, forming Dataset $P$ . Dataset $P$ has a range of $18$ inches and an interquartile range (IQR) of $8$ inches. A second dataset, Dataset $Q$ , is created by multiplying each rainfall total in Dataset $P$ by $1.2$ and then subtracting $3$ inches. What is the sum of the range and the interquartile range (IQR) of Dataset $Q$ ?

A
$25.2$
B
$26.0$
C
$28.2$
$31.2$ Answer
E
$37.2$

Answer

31.2

For any linear transformation of the form

Y = aX + b

, measures of spread such as range and interquartile range (IQR) are multiplied by

|a|

, while the constant term

b

has no effect. Multiplying the original range (

18

) and IQR (

8

) by

1.2

yields a new range of

21.6

and a new IQR of

9.6

. Summing these values gives

21.6 + 9.6 = 31.2

Step-by-Step Solution

Determine the impact of a linear transformation

Y = aX + b

on measures of dispersion.

Measures of dispersion (Range, IQR, Standard Deviation) scale by

|a|

and are unaffected by the additive constant

b

Adding or subtracting a constant shifts all data points by the same amount, leaving the distance between points unchanged, whereas multiplying by a factor scales all distances between points.

Calculate the range of Dataset

Q

\text{Range}_Q = 1.2 \times \text{Range}_P = 1.2 \times 18 = 21.6

The range scales by the multiplier

1.2

and is not affected by subtracting

3

Calculate the interquartile range (IQR) of Dataset

Q

\text{IQR}_Q = 1.2 \times \text{IQR}_P = 1.2 \times 8 = 9.6

The IQR scales by the multiplier

1.2

and is not affected by subtracting

3

Compute the sum of the range and the IQR of Dataset

Q

\text{Sum} = 21.6 + 9.6 = 31.2

Adding the newly calculated Range and IQR gives the required total spread measure.

Key Concept

Linear Transformations on Dispersion Metrics

Estimated Time:1m 30s

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