Question

the 159 heights of males from a data set of body measurements vary from a low of 151.0 cm to a high of 193.1 cm. use the range rule of thumb to estimate the standard deviation s and compare the result to the standard deviation of 9.94 cm calculated using the 159 heights. what does the result suggest about the accuracy of estimates of s found using the range rule of thumb? assume the estimate is accurate if it is within 1.4 cm.

use the range rule of thumb to estimate the standard deviation s.

( s = 10.53 ) cm

(type an integer or decimal rounded to two decimal places as needed.)

the approximation accurate because the error of the range rule of thumbs approximation is 1.4 cm.

Explanation:

Step1: Calculate the range

The range \(R\) is \(R=\text{high}-\text{low}=193.1 - 151.0=42.1\) cm.

Step2: Apply the range - rule - of - thumb formula

The range - rule - of - thumb formula for estimating the standard deviation \(s\) is \(s\approx\frac{R}{4}\).
Substitute \(R = 42.1\) into the formula: \(s\approx\frac{42.1}{4}=10.525\approx10.53\) cm.

Step3: Calculate the error

The error \(E\) is \(E=\vert\text{estimated }s-\text{calculated }s\vert\).
The calculated \(s = 9.94\) cm. So \(E=\vert10.53 - 9.94\vert=0.59\) cm.

Answer:

The approximation is accurate because the error of the range rule of thumb's approximation is less than \(1.4\) cm.