1. In the Western United States there is a saying, "Whisky is for drinken but water is for fighten!" Water and water rights have been fought over ever since ranchers and settlers moved into Wyoming, Montana, and the West. Even today farmers and ranchers fight cities and large developments over water from snowmelt that originates deep in the Rocky Mountains. A river starts in a massive and beautiful lake. Then it flows through prime trout fishing areas to a famous waterfall. After it leaves the park, the river is an important source of water for wildlife, ranchers, farmers, and cities downstream. How much water does leave the park each year? The annual flow of the river (units 10⁸ cubic meters) is shown here for 19 recent years.

25.9	32.4	33.1	19.1	17.5	24.9	21.0	45.1	30.8	34.3
27.1	29.1	25.6	31.3	23.7	24.1	23.9	25.9	18.6

(c) Find the range and standard deviation of annual flow. (Round your answers to two decimal places.)

range
standard deviation

(d) Find a 75% Chebyshev interval around the mean. (Round your answers to two decimal places.)

Lower Limit
Upper Limit

(e) Give a five-number summary of annual water flow from the river.

min	Q1	median	Q3	max

Interpret the five-number summary and the box-and-whisker plot. Where does the middle portion of the data lie?The middle portion of the data are found to have an annual flow between a lower value of and a higher value of .The total spread of annual flows goes from a lower value of to a higher value of . What is the interquartile range? Can you find data outliers? (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

(f) Another river is a smaller but very important source of water flowing out of the park from a different drainage. Ten recent years of annual water flow data are shown below (units10⁸cubicmeters).

3.83

3.81

4.01

4.84

5.81

5.50

4.31

5.81

4.31

4.67

Although smaller, is the new river more reliable? Use the coefficient of variation to make an estimate. Round your answers to two decimal place.)

original river's coefficient of variation	%
smaller river's coefficient of variation

2. Do bonds reduce the overall risk of an investment portfolio? Let x be a random variable representing annual percent return for Vanguard Total Stock Index (all stocks). Let y be a random variable representing annual return for Vanguard Balanced Index (60% stock and 40% bond). For the past several years, we have the following data.

x:	36	0	12	37	18	17	15	20	11	19
y:	26	8	26	21	24	12	21	10	4	1

(a) Computex,x²,y,y².

x		x2
y		y2

(b) Use the results of part (a) to compute the sample mean, variance, and standard deviation forxand fory. (Round your answers to four decimal places.)

	x	y
x
s2
s

(c) Compute a 75% Chebyshev interval around the mean forxvalues and also foryvalues. (Round your answers to two decimal places.)

	x	y
Lower Limit
Upper Limit

Use the intervals to compare the two funds.

75% of the returns for the balanced fund fall within a narrower range than those of the stock fund.75% of the returns for the stock fund fall within a narrower range than those of the balanced fund. 25% of the returns for the balanced fund fall within a narrower range than those of the stock fund.25% of the returns for the stock fund fall within a wider range than those of the balanced fund.

(d) Compute the coefficient of variation for each fund. (Round your answers to the nearest whole number.)

	x	y
CV	%	%

Use the coefficients of variation to compare the two funds.

For each unit of return, the stock fund has lower risk. For each unit of return, the balanced fund has lower risk. For each unit of return, the funds have equal risk.

Ifsrepresents risks andxrepresents expected return, thens/xcan be thought of as a measure of risk per unit of expected return. In this case, why is a smallerCVbetter? Explain.

A smallerCVis better because it indicates a higher risk per unit of expected return. A smallerCVis better because it indicates a lower risk per unit of expected return.

3. In this problem, we explore the effect on the standard deviation of multiplying each data value in a data set by the same constant. Consider the data set 16, 4, 8, 13, 14. (a) Use the defining formula, the computation formula, or a calculator to computes. (Round your answer to four decimal places.) s= (b) Multiply each data value by7to obtain the new data set112,28,56,91,98. Computes. (Round your answer to four decimal places.) s= (c) Compare the results of parts (a) and (b). In general, how does the standard deviation change if each data value is multiplied by a constantc?

Multiplying each data value by the same constantcresults in the standard deviation being |c| times smaller. Multiplying each data value by the same constantcresults in the standard deviation remaining the same. Multiplying each data value by the same constantcresults in the standard deviation increasing bycunits. Multiplying each data value by the same constantcresults in the standard deviation being |c| times as large.

(d) You recorded the weekly distances you bicycled in miles and computed the standard deviation to bes=4miles. Your friend wants to know the standard deviation in kilometers. Do you need to redo all the calculations?

Yes or No

Given 1 mile1.6 kilometers, what is the standard deviation in kilometers? (Enter your answer to two decimal places.) s= km

4. In this problem, we explore the effect on the standard deviation of adding the same constant to each data value in a data set. Consider the following data set. 8, 6, 14, 16, 14

(a)Use the defining formula, the computation formula, or a calculator to computes. (Enter your answer to four decimal places.)

(b)Add3to each data value to get the new data set11,9,17,19,17. Computes. (Enter your answer to four decimal places.)

(c)Compare the results of parts (a) and (b). In general, how do you think the standard deviation of a data set changes if the same constant is added to each data value?

Adding the same constantcto each data value results in the standard deviation remaining the same.

Adding the same constantcto each data value results in the standard deviation increasing bycunits. Adding the same constantcto each data value results in the standard deviation decreasing bycunits.

There is no distinct pattern when the same constant is added to each data value in a set.