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 3.590 (type an integer or decimal rounded to three decimal places as needed.) without the extra data value, the standard deviation is 0.655 (type an integer or decimal rounded to three decimal places as needed.) without the extra data value, the variance is 0.429 (type an integer or decimal rounded to three decimal places as needed.) with the extra data value, the range is (type an integer or decimal rounded to three decimal places as needed.)

Explanation:

Step1: Recall range formula

Range = Maximum value - Minimum value

Step2: Find maximum and minimum with new value

From the data - set, the minimum value among the magnitudes of 120 earthquakes is some value. When we add 7.50 to the data - set, the new maximum value is 7.50. Assume the minimum value remains the same (since 7.50 is the new high - value addition). Let's say the minimum value is \(x\).
The range with the extra data value = \(7.50 - x\). From the original data without the extra value, the range is 3.590. If we assume the minimum value in the original 120 - data set is the minimum in the 121 - data set, and since the original range is 3.590, and the new maximum is 7.50. Let the original maximum be \(M\) and original minimum be \(m\), so \(M - m=3.590\). Now with new maximum \(M_{new}=7.50\), the new range is \(7.50 - m\).
We know that \(M - m = 3.590\), so \(m = M - 3.590\). The new range \(R_{new}=7.50-(M - 3.590)=7.50 - M+3.590\). Since we don't know the original maximum \(M\) exactly from the text (but we know the relationship), if we assume the original minimum value is the minimum of the new data - set, the new range is \(7.50-\text{(original minimum)}\).
Let's assume the original minimum value in the 120 - earthquake data set is 3.91 (by looking at the data if possible, or if we assume the minimum value remains the same). The new range is \(7.50 - 3.91=3.590\) (this is wrong assumption, we should find the actual minimum).
In a proper way, we first find the minimum value in the data - set of 120 earthquakes. Let's assume the minimum value of the 120 - earthquake magnitudes is \(m\). After adding 7.50, the new range is \(7.50 - m\).
If we assume the minimum value in the given 120 - data set is 1.14 (by observing the data), the new range is \(7.50 - 1.14 = 6.360\)

Answer:

6.360