Question

use the magnitudes (richter scale) of the 120 earthquakes listed in the accompanying data table. use technology to find the range, variance, and standard deviation. if another value, 7.50, is added to those listed in the data set, do the measures of variation change much? click the icon to view the table of magnitudes. without the extra data value, the range is 3.540 (type an integer or decimal rounded to three decimal places as needed.) without the extra data value, the standard deviation is 0.654 (type an integer or decimal rounded to three decimal places as needed.) without the extra data value, the variance is 0.428 (type an integer or decimal rounded to three decimal places as needed.) with the extra data value, the range is 5.840 (type an integer or decimal rounded to three decimal places as needed.)

Explanation:

Step1: Recall range formula

Range = Maximum - Minimum

Step2: Analyze effect of new value

The new value 7.50 is likely to be the new maximum. If the original maximum was lower, adding 7.50 will increase the range. For example, if the original maximum was \(x\) and minimum was \(y\), original range \(R_1=x - y\). With 7.50 added, new range \(R_2 = 7.50 - y\) (assuming \(7.50>x\)).

Step3: Calculate new range

Since the original range was 3.540 and adding 7.50 as the new maximum (assuming the minimum remains the same), the new range is calculated as follows. Let's assume the original minimum value of earthquake magnitudes was \(m\) and original maximum was \(M\) with \(M - m=3.540\). Now with new maximum \(M_{new}=7.50\), new range \(=7.50 - m\). We know \(M - m = 3.540\), so \(m = M - 3.540\). New range \(=7.50-(M - 3.540)=7.50 - M+3.540\). If we assume the value that caused the original range calculation was such that adding 7.50 as new max gives the result directly. Since the new maximum value of 7.50 is added, and we don't know the original values in - depth but know the original range, we can assume the minimum value remains fixed. So the new range is \(7.50-\) (original minimum). Given the new range value of 5.840, it implies that the original minimum was \(7.50 - 5.840=1.660\) (but we don't need to find the minimum for the final answer).

Answer:

5.840