Chapter 3, Problem 9CRP

a.

To determine

Check whether there is a guaranteed amount of water the farmers, ranchers, and cities will get from the Yellowstone River each year.

Answer to Problem 9CRP

No, there is no guaranteed amount of water the farmers, ranchers, and cities get from the Yellowstone River each year.

Explanation of Solution

The important source of water for wildlife, rancher, farmers, and cities downstream is Yellowstone River. The data show that the annual flow for recent 19 years of the Yellowstone River, which does not show any guarantee that they give water for each year.

The annual flows are random variables. Hence, there is no guaranteed amount of water the farmers, ranchers, and cities will get from the Yellowstone River each year.

b.

To determine

Find the expected annual flow from the Yellowstone snowmelt.

Find the mean, median, and mode.

Answer to Problem 9CRP

The expected annual flow from the Yellowstone snowmelt is 27.05.

The value of mean is 27.05.

The value of median is 25.9.

The value of mode is 25.9.

Explanation of Solution

Step-by-step procedure to obtain the mean, median, and mode using the MINITAB software:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns Annual flow.
• Check Options, Select Mean, Median and Mode.
• Click OK in all dialogue boxes.

The output obtained using the MINITAB software is given below:

From the MINITAB output, the value of mean is 27.05, the value of median is 25.9, and the value of mode is 25.9.

c.

To determine

Find the range and standard deviation of annual flow.

Answer to Problem 9CRP

The range is 27.6.

The value of standard deviation is 6.61.

Explanation of Solution

Step-by-step procedure to obtain the range and standard deviation using the MINITAB software:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns Annual flow.
• Check Options, Select Mean, Median and Mode.
• Click OK in all dialogue boxes.

The output obtained using the MINITAB software is given below:

From the MINITAB output, the range is 27.6 and the value of standard deviation is 6.61.

d.

To determine

Find the 75% Chebyshev interval around the mean.

Answer to Problem 9CRP

The 75% Chebyshev interval around the mean is 13.83 and 40.27.

Explanation of Solution

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

$\begin{array}{c}\overline{x}-2s=27.05-2\left(6.61\right)\\ =27.05-13.22\\ =13.83\\ \overline{x}+2s=27.05+2\left(6.61\right)\\ =27.05+13.22\\ =40.27\end{array}$

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

e.

To determine

Find the five-number summary of the annual water flow.

Draw the box-and-whisker plot.

Interpret the five-number summary and box-and-whisker plot.

Find the values where the middle portion of the data lies.

Find the interquartile range.

Identify the data outliers.

Answer to Problem 9CRP

The five-number summary of the annual water flow is 17.5, 23.7, 25.9, 31.8, and 45.1.

The box-and-whisker plot is shown below:

The middle portion of the data lies between 23.7 and 31.8.

The value of interquartile range is 8.1.

The value 45.1 is an outlier.

Explanation of Solution

Step-by-step procedure to obtain the five-number summary of the annual water flow using MINITAB software:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns Annual flow.
• Check Options, Select Minimum, Maximum, first quartile, Median, third quartile and IQR.
• Click OK in all dialogue boxes.

The output obtained using the MINITAB software is given below:

From the MINITAB output, the five-number summary of annual water flow is 17.5, 23.7, 25.9, 31.8, and 45.1.

Step-by-step procedure to draw the box-and-whisker plot using the MINITAB software:

• Choose Graph > Boxplot or Stat > EDA > Boxplot.
• Under One Y, choose Simple. Click OK.
• In Graph variables, enter the data of Annual flow.
• Click OK in all dialogue boxes.

Interpretation:

From the box-and-whisker plot, it is observed that the distribution of the annual water flow is skewed to the right.

From the output, it is observed that the middle portion of the data lies between 23.7 and 31.8.

From the MINITAB output, the value of interquartile range is 8.1.

From the box-and-whisker plot, it is observed that the value 45.1 is an outlier.

f.

To determine

Check whether the Madison is more reliable using the coefficient of variation.

Explanation of Solution

Step-by-step procedure to obtain the coefficient of variation using the MINITAB software:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns Yellowstone and Madison.
• Check Options, Select coefficient of variation.
• Click OK in all dialogue boxes.

The output obtained using the MINITAB software is given below:

From the MINITAB output, the coefficient of variation for the annual water flow of Yellowstone is 24.43, and the coefficient of variation for the annual water flow of Madison is 16.6.

From the result, it is observed that the coefficient of variation for the water flow of Madison is smaller when compared to the coefficient of variation for the annual water flow of Yellowstone. This indicates that the spread of river flow is smaller for Madison river. Hence, the Madison flower is more consistent.

g.

To determine

Check whether it is safe to allocate at least 27 units of Yellowstone River water each year for agricultural and domestic use.

Explanation of Solution

From the results, it is observed that the median water flow of Yellowstone River is 25.9, which indicates that more than half of the river flows are below 27 units.

Hence, it is not safe to allocate at least 27 units of Yellowstone River water each year for agricultural and domestic use.