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 Finite mathematics exercises: linear and exponential regression

In Exercises 1–4 calculate the value of SSE for the given points and line.

 Points: (1, 1), (2, 2), (3, 6); Line: y = 2x - 1   SSE = Points: ; Line:   SSE = Points: (−2, 4), (0, 2), (3, −8), (4, −10); Line: y = -3x + 2   SSE = Points: ; Line:   SSE =

Find the regression line associated with the given points in Exercises 5–8. The coefficients should be rounded to four significant digits. Graph the data and the regression line.

Points: (1, 1), (2, 2), (3, 4);   Regression line: y =
Points: ; Regression line: y =
Graph the regression line. (To draw a line, click on any two points on the line you want as accurately as possible.)
Points: (0, -1), (1, 3), (4, 6), (5, 0);  Regression line: y =
Points: ; Regression line: y =
Graph the regression line. (To draw a line, click on any two points on the line you want as accurately as possible.)

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The use of technology to compute regression lines is recommended in the applications exercises below.

Pollution Control According to surveys, the percentage of new plant and equipment expenditures by US manufacturing companies allocated to pollution control in the period 1975–1987 was as shown:*

 1975 1980 1981 1984 1987 9.3 4.8 4.3 3.3 4.3

*Source: Survey of Current Business 58, 62, 66, 68

Obtain a linear regression model of the form y = mx + b where x is time in years since 1975, and use it to estimate the figure for 1985. (Round your regression coefficients to four decimal places, and the predicted value for 1985 to one decimal place.)

 Regression equation: y = Percentage for 1985: (one decimal place)

Pollution Control The percentage of new plant and equipment expenditures by US public utility companies on pollution control is approximately as shown:**

 1975 1980 1981 1984 1987

**Figures are approximate (partially randomized). Source: Survey of Current Business 58, 62, 66, 68

Obtain a linear regression model of the form y = mx + b where x is time in years since 1970, and use it to estimate the figure for 1988. (Round your regression coefficients to four decimal places, and the predicted value for 1988 to one decimal place.)

 Regression equation: y = Percentage for 1988: (one decimal place)

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Soybean Production: Brazil The following table shows soybean production, in millions of tons, in Brazil's Cerrados region, as a function of the cultivated area, in millions of acres.

 Area(Millions of Acres) 25 30 32 40 52 Production(Millions of Tons) 15 25 30 40 60

*Source: Brazil Agriculture Ministry/New York Times, December 12, 2004, p. N32.

(a) Use technology to obtain the equation of the regression line with x = area and y = production. Coefficients should be rounded to at least two decimal places.)

 Regression equation: y =

(b) The regression equation tells you that soybean production

Soybean Production: Mars The following table shows soybean production, in millions of kilos, in the recently cultivated foothills of Olympus Mons, as a function of the cultivated area, in thousands of square meters.

 Area(Thousand sq. meters) Production(Million kilos)

*Source: Mars Imperial Ministry of Food/Daily Planet, Aug. 12, 2497, p. 478.

(a) Use technology to obtain the equation of the regression line with x = area and y = production. Coefficients should be rounded to two decimal places.)

 Regression equation: y =

(b) The regression equation tells you that soybean production

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Profit The following chart shows the net income, in millions of dollars, of the Disney Company for the years 1984-1992.

Profits estimated to the nearest \$5 million.
Source: Company Reports/The New York Times, December, 1992.

(a) Use technology to find a least squares fit linear model for this data. (Find profit p as a function of the year t, with t = 0 corresponding to 1980. Coefficients should be rounded to the nearest whole number.)

p =

(b) According to the model, how fast, to the nearest million dollars per year, was Disney's profit increasing over the period?

\$per year

Stock Prices Repeat the previous exercise, but this time model Walt Disney's stock prices as shown in the following chart:

#Stock prices are rough approximations.
Source: Company Reports/The New York Times, December, 1992.

(a) Use technology to find a least squares fit linear model for this data. (Find stock price p as a function of the year t, with t = 0 corresponding to 1980. Coefficients should be rounded to the nearest whole number.)

p =

(b) According to the model, how fast was Disney stock price increasing over the period?

\$per year

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Power Usage Here are some data and a graph showing monthly electricity use versus the size of a home:

 Size (thousand sq. ft) \pmb{x} 1.29 1.35 1.47 1.6 1.71 1.84 1.98 2.23 2.4 2.93 Usage (kw-hrs) \pmb{y} 1180 1170 1260 1490 1570 1710 1800 1840 1960 1950

(a) Use technology to obtain the equation of the regression line with coefficients rounded to one decimal place.

Regression line: y =
(b) Graph the regression line by clicking on two points on the graph and pressing "Check graph" below the graph.
(c) The graphed data suggests a curve rather than a line. Which of the following kinds of curve will best fit the data?:

Housing Costs: Mars The cost of housing in Utarek, Mars, goes up rapidly with size, due to the extreme limitations of space. The following table shows the cost of housing in zonars () for various size dwellings.

 Size x (hundred sq. m) Price y ( million)

(a) Use technology to obtain the equation of the regression line with coefficients rounded to one decimal place.

Regression line: y =
(b) Graph the regression line by clicking on two points on the graph and pressing "Check" below the graph.
(c) The graphed data suggests a curve rather than a line. Whis of the following kinds of curve will best fit the data?:

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Find the regression exponential curve y = Ab^x associated with the given points in Exercises 1618. The coefficients should be rounded to four significant digits. Graph the data and the regression curve.

Points: (1, 1), (2, 2), (3, 4);   Exponential curve: y =
Points: ; Exponential curve: y =
Graph the regression curve. (To draw an exponential curve, click on any two points on the curve you want as accurately as possible.)
Points: (0, 1), (1, 3), (4, 6), (5, 1);  Exponential curve: y =
Points: ; Exponential curve: y =
Graph the exponenial curve. (To draw an exponential curve, click on any two points on the curve you want as accurately as possible.)

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Big Brother In 1995, the FBI was seeking the ability to monitor 74,250 phone lines at once. The following chart shows the number of phone lines monitored from 1987 through 1993.

Source: Electronic Privacy Information Center, Justice Department, Administrative Office of the United States Courts/The New York Times, November 2, 1995, p. D5.

Use an exponential model of this data y = Ab^x with x the year since 1987 to project, to the nearest 100, the number of phone lines tapped by the FBI in 1994. (Round your answer to the nearest 100 phone lines.)

Exponential curve: y =

Projected value in 1994:

Beer The following chart shows approximate after-tax profits, in millions of dollars, of South African Breweries for the years 1991 through 1997.

Data are rounded and randomized by up to ± \$30 million. Source: Comapny reports/Bloomberg Financial Markets/The New York Times, August 27, 1997.

Use an exponential model of this data y = Ab^x with x the year since 1990 and coefficients rounded to two decimal places to predict SAB's profit in 2000. (Round your answer to the nearest \$10 million.)

Exponential curve: y =

Projected value in 2000:
Graph the exponenial curve. (To draw an exponential curve, click on any two points on the curve you want as accurately as possible and press "Check Graph".)

Last Updated: January 2008