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, 7.00, is added to those listed in this 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 (type an integer or decimal rounded to three decimal places as needed.) without the extra data value, the standard deviation is (type an integer or decimal rounded to three decimal places as needed.) without the extra data value, the variance is (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.) with the extra data value, the standard deviation is (type an integer or decimal rounded to three decimal places as needed.) with the extra data value, the variance is (type an integer or decimal rounded to three decimal places as needed.) do the measures of variation change much with the extra data value? choose the correct answer below. the ranges are the variations are and the standard deviations are

Explanation:

Step1: Recall range formula

Range = Max - Min

Step2: Recall variance formula

$s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}$, where $x_{i}$ are data - points, $\bar{x}$ is the mean, and $n$ is the number of data - points

Step3: Recall standard deviation formula

$s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$
Since the data table is not provided, we assume we have a set of data values $x_1,x_2,\cdots,x_{120}$ for the first part and $x_1,x_2,\cdots,x_{120},7.00$ for the second part.

1. Without the extra data value:

• First, find the maximum ($Max_1$) and minimum ($Min_1$) of the 120 - data set to calculate the range $R_1=Max_1 - Min_1$.
• Calculate the mean $\bar{x}_1=\frac{\sum_{i = 1}^{120}x_{i}}{120}$.
• Then calculate the variance $s_1^{2}=\frac{\sum_{i = 1}^{120}(x_{i}-\bar{x}_1)^{2}}{119}$.
• And the standard deviation $s_1=\sqrt{s_1^{2}}$.

1. With the extra data value:

• Find the maximum ($Max_2$) and minimum ($Min_2$) of the 121 - data set to calculate the range $R_2=Max_2 - Min_2$.
• Calculate the mean $\bar{x}_2=\frac{\sum_{i = 1}^{120}x_{i}+7.00}{121}$.
• Then calculate the variance $s_2^{2}=\frac{\sum_{i = 1}^{120}(x_{i}-\bar{x}_2)^{2}+(7.00 - \bar{x}_2)^{2}}{120}$.
• And the standard deviation $s_2=\sqrt{s_2^{2}}$.
• To determine if the measures of variation change much, compare $R_1$ and $R_2$, $s_1^{2}$ and $s_2^{2}$, $s_1$ and $s_2$. If the differences are large, we say they change much; if the differences are small, we say they do not change much.

Since we don't have the actual data, we can't give numerical answers. But the general steps to solve this problem are as above.

Answer:

Without the extra data value:
Range: [To be calculated using actual data]
Standard deviation: [To be calculated using actual data]
Variance: [To be calculated using actual data]
With the extra data value:
Range: [To be calculated using actual data]
Standard deviation: [To be calculated using actual data]
Variance: [To be calculated using actual data]
Do the measures of variation change much: [To be determined by comparing the calculated values]