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 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 . (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

Explanation:

Step1: Recall range formula

Range = Maximum - Minimum

Step2: Find maximum and minimum without extra value

Sort the 120 - earthquake magnitudes data set. Let the maximum be $M_{max}$ and minimum be $M_{min}$. Calculate Range$_{1}=M_{max}-M_{min}$

Step3: Recall variance formula

The sample - variance formula is $s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}$, where $x_{i}$ are the data points, $\bar{x}$ is the sample mean, and $n$ is the number of data points. First, calculate the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, then calculate $(x_{i}-\bar{x})^{2}$ for each $i$, sum them up, and divide by $n - 1$ to get $s_{1}^{2}$

Step4: Recall standard - deviation formula

The sample standard deviation $s=\sqrt{s^{2}}$. So, $s_{1}=\sqrt{s_{1}^{2}}$

Step5: Add the extra value (7.00)

Add 7.00 to the data set. Now $n$ becomes 121.

Step6: Recalculate range

Find the new maximum $M_{new - max}$ and new minimum $M_{new - min}$ in the 121 - data - point set. Calculate Range$_{2}=M_{new - max}-M_{new - min}$

Step7: Recalculate variance

Recalculate the mean $\bar{x}_{new}=\frac{\sum_{i = 1}^{121}x_{i}}{121}$, then calculate $(x_{i}-\bar{x}_{new})^{2}$ for each $i$, sum them up, and divide by $121 - 1$ to get $s_{2}^{2}$

Step8: Recalculate standard - deviation

Calculate $s_{2}=\sqrt{s_{2}^{2}}$

Step9: Analyze change

Compare Range$_{1}$ and Range$_{2}$, $s_{1}^{2}$ and $s_{2}^{2}$, $s_{1}$ and $s_{2}$ to determine if the measures of variation change much.

Answer:

Without performing the actual calculations using the data set (as it is a large set and best done with software like Excel, R, or Python), we cannot provide the numerical values. But the steps above show how to calculate the range, variance, and standard deviation without and with the extra data value and analyze the change. If you use software:

1. In Excel, you can use functions like MAX, MIN, STDEV.S, and VAR.S to calculate the range, standard deviation, and variance respectively. First, calculate these for the 120 - data - point set, then add the value 7.00 to the data set and recalculate.
2. In R, you can use functions like max, min, sd, and var. For example, if your data set is named 'data', you can calculate the range as max(data)-min(data), the variance as var(data), and the standard deviation as sd(data). Then add the new value to the data set and recalculate.
3. In Python, you can use libraries like numpy. If your data is in a numpy array named 'arr', you can calculate the range as np.max(arr)-np.min(arr), the variance as np.var(arr, ddof = 1), and the standard deviation as np.std(arr, ddof = 1). Then add the new value to the array and recalculate.