In a meteorological study, the daily peak wind speeds (in kilometers per hour) recorded ov… — GRE General Test

In a meteorological study, the daily peak wind speeds (in kilometers per hour) recorded over 7 consecutive days were $14, 18, 21, 24, 27, 31,$ and $37$ . As part of a data calibration process, each recorded wind speed is increased by $15\%$ and then increased by an additional constant of $5\text{ km/h}$ . What is the range, in kilometers per hour, of the calibrated daily peak wind speeds?

Answer: 26.45 km/h

Answer

The range of the calibrated daily peak wind speeds is 26.45 km/h.

For any dataset transformed linearly according to

y = a \cdot x + b

where

a > 0

, the range of the transformed dataset is given by

\text{Range}(Y) = a \cdot \text{Range}(X)

. Here, increasing each value by

15\%

corresponds to

a = 1 + 0.15 = 1.15

, and adding

5

corresponds to

b = 5

. The original maximum is

37

and the original minimum is

14

, giving an original range of

37 - 14 = 23

. Therefore, the new range is

1.15 \times 23 = 26.45\text{ km/h}

. The constant shift of

5\text{ km/h}

affects the center (mean, median) but has zero effect on the dispersion.

Step-by-Step Solution

Determine the range of the original set of wind speeds.

Original Range = 37 - 14 = 23 km/h.

The range is defined as the difference between the maximum and minimum values in a dataset.

Apply the properties of linear transformations to measures of dispersion.

The transformation is y = 1.15x + 5. The range is scaled by 1.15 and unaffected by the addition of 5.

Adding a constant shift to all data points shifts the entire distribution without changing the spread (dispersion), whereas multiplying all data points by a constant factor 'a' scales all measures of dispersion by |a|.

Compute the calibrated range.

Calibrated Range = 1.15 * 23 = 26.45 km/h.

Multiplying the original range of 23 by the scale factor of 1.15 gives the exact range of the transformed dataset.

Key Concept

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

Estimated Time:1m 30s

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