Chapter 3.2, Problem 18P

a.

To determine

Find the values ${\displaystyle \sum x},{\displaystyle \sum x^2},{\displaystyle \sum y},\text{ and }{\displaystyle \sum y^2}$ using calculator.

Answer to Problem 18P

The values of ${\displaystyle \sum x}=\underset{\_}{284.95}$ and ${\displaystyle \sum x^2}=\underset{\_}{7,046.80}$.

The values of ${\displaystyle \sum y}=\underset{\_}{421.5}$ and ${\displaystyle \sum y^2}=\underset{\_}{14,562.29}$.

Explanation of Solution

Step-by-step procedure to verify ${\displaystyle \sum x}$ and ${\displaystyle \sum x^2}$ using Ti83 calculator is given below:

• Press STAT.
• Select Edit.
• Enter the values in L1.
• Press STAT and Choose CALC.
• Select 1-Var Stats.
• To select the variable L1, Press 2-nd, and then press 1.
• Press Enter.

Output obtained using the Ti83 calculator is given below:

From the output, the values of ${\displaystyle \sum x}=284.95$ and ${\displaystyle \sum x^2}=7,046.80$.

Step-by-step procedure to verify ${\displaystyle \sum y}$ and ${\displaystyle \sum y^2}$ using Ti83 calculator is given below:

• Press STAT.
• Select Edit.
• Enter the values in L2.
• Press STAT and Choose CALC.
• Select 1-Var Stats.
• To select the variable L2, Press 2-nd, and then press 2.
• Press Enter.

Output obtained using the Ti83 calculator is given below:

From the output, the values of ${\displaystyle \sum y}=421.5$ and ${\displaystyle \sum y^2}=14,562.29$.

b.

To determine

Find the sample mean, variance, and standard deviation for x and y using the computation formula.

Answer to Problem 18P

The values of sample mean, variance, and standard deviation for x using the computation formula is 20.35, 96 and 9.79.

The values of sample mean, variance, and standard deviation for y using the computation formula is 28.1, 194 and 13.93.

Explanation of Solution

The sample variance using the computation formula is as follows:

$s^2=\frac{{\displaystyle \sum x^2-\frac{{\left({\displaystyle \sum x}\right)}^2}{n}}}{n-1}$

Where, $\overline{x}$ is the mean, n is the number of values in the data, and x is the values in the data.

The sample standard deviation using the computation formula is as follows:

$s=\sqrt{\frac{{\displaystyle \sum x^2-\frac{{\left({\displaystyle \sum x}\right)}^2}{n}}}{n-1}}$

Where, $\overline{x}$ is the mean, n is the number of values in the data, and x is the values in the data.

The sample mean for x is obtained below:

$\begin{array}{c}\overline{x}=\frac{{\displaystyle \sum x}}{n}\\ =\frac{284.95}{14}\\ =20.35\end{array}$

Thus, the sample mean is 20.35.

The sample variance for x using the computation formula is obtained below:

$\begin{array}{c}{s}^2=\frac{7046.80-\frac{{\left(284.95\right)}^2}{14}}{14-1}\\ =96\end{array}$

Thus, the sample variance for x using the computation formula is 96.

The sample standard deviation for x using the computation formula is obtained below:

$\begin{array}{c}s=\sqrt{96}\\ =9.79\end{array}$

Thus, the sample standard deviation for x using the computation formula is 9.79.

The sample mean for y is obtained below:

$\begin{array}{c}\overline{y}=\frac{{\displaystyle \sum y}}{n}\\ =\frac{421.5}{15}\\ =28.1\end{array}$

Thus, the sample mean is 28.1.

The sample variance for y using the computation formula is obtained below:

$\begin{array}{c}{s}^2=\frac{14562.29-\frac{{\left(421.5\right)}^2}{15}}{15-1}\\ =194.1\\ \approx 194\end{array}$

Thus, the sample variance for y using the computation formula is 194.

The sample standard deviation for y using the computation formula is obtained below:

$\begin{array}{c}s=\sqrt{194}\\ =13.93\end{array}$

Thus, the sample standard deviation for y using the computation formula is 13.93.

c.

To determine

Find the 75% Chebyshev interval around the mean for each fund.

Compare the two grids.

Answer to Problem 18P

The 75% Chebyshev interval around the mean for the Grid E is 0.77 and 39.93.

The 75% Chebyshev interval around the mean for the Grid H is 0.24 and 55.96.

Explanation of Solution

The 75% Chebyshev interval around the mean for the Grid Eis obtained below:

$\begin{array}{c}\overline{x}-2s=20.35-2\left(9.79\right)\\ =0.77\\ \overline{x}+2s=20.35+2\left(9.79\right)\\ =39.93\end{array}$

Thus, the 75% Chebyshev interval around the mean for the Grid E is 0.77 and 39.93.

The 75% Chebyshev interval around the mean for the Grid H is obtained below:

$\begin{array}{c}\overline{y}-2s=28.1-2\left(13.93\right)\\ =0.24\\ \overline{y}+2s=28.1+2\left(13.93\right)\\ =55.96\end{array}$

Thus, the 75% Chebyshev interval around the mean for the Grid H is 0.24 and 55.96.

From the result, it is observed that the 75% Chebyshev interval around the mean for the Grid H is wider when compared to the Grid E.

d.

To determine

Find and compare the coefficient of variation for x and y.

Answer to Problem 18P

The coefficient of variation for x is 48%.

The coefficient of variation for y is 50%.

Explanation of Solution

The formula for the coefficient of variation is as follows:

$CV=\frac{s}{\overline{x}}\times 100\%$

Where, s is the sample standard deviation and $\overline{x}$ is the sample mean.

The coefficient of variation for x is obtained below:

$\begin{array}{c}CV=\frac{9.79}{20.35}\times 100\%\\ =48\%\end{array}$

Thus, the coefficient of variation for x is 48%.

The coefficient of variation for y is obtained below:

$\begin{array}{c}CV=\frac{13.93}{28.1}\times 100\%\\ =50\%\end{array}$

Thus, the coefficient of variation for y is 50%.

From the results, the coefficient of variation for the Grid E is less when compared to the coefficient of variation for the Grid H. Hence, the Grid H has more buried artifacts.