A quality control engineer records the thickness, in millimeters, of 9 sample components: … — GRE General Test

A quality control engineer records the thickness, in millimeters, of 9 sample components: $11, 13, 15, 17, 19, 21, 23, 25,$ and $27$ . Each thickness measurement $x$ is then converted to a scaled rating $y$ using the linear formula $y = 1.5x + 4.8$ . What is the interquartile range (IQR) of the transformed dataset of $y$ -values?

Answer: 15

Answer

The interquartile range of the transformed dataset is 15.

The interquartile range (IQR) measures the spread of the middle 50% of the data (

Q_3 - Q_1

). For the original dataset

11, 13, 15, 17, 19, 21, 23, 25, 27

, the median is

19

Q_1

is the median of the lower half

\{11, 13, 15, 17\}

, which is

\frac{13+15}{2} = 14

Q_3

is the median of the upper half

\{21, 23, 25, 27\}

, which is

\frac{23+25}{2} = 24

. Thus, the original

\text{IQR} = 24 - 14 = 10

. Under a linear transformation

y = ax + b

, measures of position shift by

a x + b

, so

Q_1(y) = 1.5(14) + 4.8 = 25.8

and

Q_3(y) = 1.5(24) + 4.8 = 40.8

. Subtracting these yields

\text{IQR}(y) = 40.8 - 25.8 = 15

. Notice that this is simply

1.5 \times 10

, as adding a constant shifts the location of the distribution but leaves measures of spread unchanged.

Step-by-Step Solution

Find the quartiles of the original 9-element dataset.

Q_1 = 14

and

Q_3 = 24

The median of the dataset is 19 (the 5th value). The lower half of the data consists of

11, 13, 15, 17

, so

Q_1 = \frac{13 + 15}{2} = 14

. The upper half consists of

21, 23, 25, 27

, so

Q_3 = \frac{23 + 25}{2} = 24

Calculate the interquartile range of the original dataset.

\text{IQR}_x = 10

\text{IQR}_x = Q_3 - Q_1 = 24 - 14 = 10

Apply the linear transformation rules to find the transformed interquartile range.

\text{IQR}_y = 15

For a linear transformation

y = ax + b

, the interquartile range scales by

|a|

, so

\text{IQR}_y = |a| \cdot \text{IQR}_x = 1.5 \times 10 = 15

. The constant addition of

4.8

shifts all values equally and does not affect the spread/IQR.

Key Concept

Effect of linear transformations on measures of dispersion (IQR, standard deviation, range)

Alternative Method

Transform the individual quartiles directly:

Q_1(y) = 1.5(14) + 4.8 = 25.8

and

Q_3(y) = 1.5(24) + 4.8 = 40.8

. Then calculate the new IQR directly as

40.8 - 25.8 = 15

Estimated Time:1m 30s

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Key Concept

Common Mistakes

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