Section Exercises
1. Describe what it means if there is a model breakdown when using a linear model.
2. What is interpolation when using a linear model?
3. What is extrapolation when using a linear model?
4. Explain the difference between a positive and a negative correlation coefficient.
5. Explain how to interpret the absolute value of a correlation coefficient.
6. A regression was run to determine whether there is a relationship between hours of TV watched per day (x) and number of sit-ups a person can do (y). The results of the regression are given below. Use this to predict the number of sit-ups a person who watches 11 hours of TV can do.
[latex]\begin{cases}y=ax+b\\ a=-1.341\\ b=32.234\\ r=-0.896\end{cases}\\[/latex]
7. A regression was run to determine whether there is a relationship between the diameter of a tree (x, in inches) and the tree’s age (y, in years). The results of the regression are given below. Use this to predict the age of a tree with diameter 10 inches.
[latex]\begin{cases}y=ax+b\\ a=6.301\\ b=-1.044\\ r=-0.970\end{cases}\\[/latex]
For the following exercises, draw a scatter plot for the data provided. Does the data appear to be linearly related?
8.
| 0 |
2 |
4 |
6 |
8 |
10 |
| –22 |
–19 |
–15 |
–11 |
–6 |
–2 |
9.
| 1 |
2 |
3 |
4 |
5 |
6 |
| 46 |
50 |
59 |
75 |
100 |
136 |
10.
| 100 |
250 |
300 |
450 |
600 |
750 |
| 12 |
12.6 |
13.1 |
14 |
14.5 |
15.2 |
11.
| 1 |
3 |
5 |
7 |
9 |
11 |
| 1 |
9 |
28 |
65 |
125 |
216 |
12. For the following data, draw a scatter plot. If we wanted to know when the population would reach 15,000, would the answer involve interpolation or extrapolation? Eyeball the line, and estimate the answer.
| Year |
Population |
| 1990 |
11,500 |
| 1995 |
12,100 |
| 2000 |
12,700 |
| 2005 |
13,000 |
| 2010 |
13,750 |
13. For the following data, draw a scatter plot. If we wanted to know when the temperature would reach 28 °F, would the answer involve interpolation or extrapolation? Eyeball the line and estimate the answer.
| Temperature, °F |
16 |
18 |
20 |
25 |
30 |
| Time, seconds |
46 |
50 |
54 |
55 |
62 |
For the following exercises, match each scatterplot with one of the four specified correlations below.
14. [latex]r=0.\text{95}\\[/latex]
15. [latex]r=-0.\text{89}\\[/latex]
16. [latex]r=0.26\\[/latex]
17. [latex]r=-0.39\\[/latex]
For the following exercises, draw a best-fit line for the plotted data.
18.
19.
20.
21.
22. The U.S. Census tracks the percentage of persons 25 years or older who are college graduates. That data for several years is given in the table below.[footnote]http://www.census.gov/hhes/socdemo/education/data/cps/historical/index.html. Accessed 5/1/2014.[/footnote] Determine whether the trend appears linear. If so, and assuming the trend continues, in what year will the percentage exceed 35%?
| Year |
Percent Graduates |
| 1990 |
21.3 |
| 1992 |
21.4 |
| 1994 |
22.2 |
| 1996 |
23.6 |
| 1998 |
24.4 |
| 2000 |
25.6 |
| 2002 |
26.7 |
| 2004 |
27.7 |
| 2006 |
28 |
| 2008 |
29.4 |
23. The U.S. import of wine (in hectoliters) for several years is given below. Determine whether the trend appears linear. If so, and assuming the trend continues, in what year will imports exceed 12,000 hectoliters?
| Year |
Imports |
| 1992 |
2665 |
| 1994 |
2688 |
| 1996 |
3565 |
| 1998 |
4129 |
| 2000 |
4584 |
| 2002 |
5655 |
| 2004 |
6549 |
| 2006 |
7950 |
| 2008 |
8487 |
| 2009 |
9462 |
24. The table below shows the year and the number of people unemployed in a particular city for several years. Determine whether the trend appears linear. If so, and assuming the trend continues, in what year will the number of unemployed reach 5?
| Year |
Number Unemployed |
| 1990 |
750 |
| 1992 |
670 |
| 1994 |
650 |
| 1996 |
605 |
| 1998 |
550 |
| 2000 |
510 |
| 2002 |
460 |
| 2004 |
420 |
| 2006 |
380 |
| 2008 |
320 |
For the following exercises, use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy.
25.
| x |
8 |
15 |
26 |
31 |
56 |
| y |
23 |
41 |
53 |
72 |
103 |
26.
| x |
5 |
7 |
10 |
12 |
15 |
| y |
4 |
12 |
17 |
22 |
24 |
27.
| x |
y |
x |
y |
| 3 |
21.9 |
11 |
15.76 |
| 4 |
22.22 |
12 |
13.68 |
| 5 |
22.74 |
13 |
14.1 |
| 6 |
22.26 |
14 |
14.02 |
| 7 |
20.78 |
15 |
11.94 |
| 8 |
17.6 |
16 |
12.76 |
| 9 |
16.52 |
17 |
11.28 |
| 10 |
18.54 |
18 |
9.1 |
28.
| x |
y |
| 4 |
44.8 |
| 5 |
43.1 |
| 6 |
38.8 |
| 7 |
39 |
| 8 |
38 |
| 9 |
32.7 |
| 10 |
30.1 |
| 11 |
29.3 |
| 12 |
27 |
| 13 |
25.8 |
29.
| x |
y |
| 21 |
17 |
| 25 |
11 |
| 30 |
2 |
| 31 |
−1 |
| 40 |
−18 |
| 50 |
−40 |
30.
| x |
y |
| 100 |
2000 |
| 80 |
1798 |
| 60 |
1589 |
| 55 |
1580 |
| 40 |
1390 |
| 20 |
1202 |
31.
| x |
y |
| 900 |
70 |
| 988 |
80 |
| 1000 |
82 |
| 1010 |
84 |
| 1200 |
105 |
| 1205 |
108 |
32. Graph [latex]f\left(x\right)=0.5x+10\\[/latex] . Pick a set of 5 ordered pairs using inputs [latex]x=\text{-2},\text{ 1},\text{ 5},\text{ 6},\text{ 9}\\[/latex] and use linear regression to verify that the function is a good fit for the data.
33. Graph [latex]f\left(x\right)=-2x - 10\\[/latex]. Pick a set of 5 ordered pairs using inputs [latex]x=\text{-2},\text{ 1},\text{ 5},\text{ 6},\text{ 9}\\[/latex] and use linear regression to verify the function.
For the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The following ordered pairs shows dollars and the number of units sold in hundreds and the profit in thousands of over the ten-year span, (number of units sold, profit) for specific recorded years:
[latex]\left(\text{46},\text{ 1},\text{6}00\right), \left(\text{48},\text{ 1},\text{55}0\right), \left(\text{5}0,\text{ 1},\text{5}0\text{5}\right), \left(\text{52},\text{ 1},\text{54}0\right), \left(\text{54},\text{ 1},\text{495}\right)\\[/latex] .
34. Use linear regression to determine a function P where the profit in thousands of dollars depends on the number of units sold in hundreds.
35. Find to the nearest tenth and interpret the x-intercept.
36. Find to the nearest tenth and interpret the y-intercept.
For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population and the year over the ten-year span, (population, year) for specific recorded years:
(2500, 2000), (2650, 2001), (3000, 2003), (3500, 2006), (4200, 2010)
37. Use linear regression to determine a function y, where the year depends on the population. Round to three decimal places of accuracy.
38. Predict when the population will hit 8,000.
For the following exercises, consider this scenario: The profit of a company increased steadily over a ten-year span. The following ordered pairs show the number of units sold in hundreds and the profit in thousands of over the ten year span, (number of units sold, profit) for specific recorded years:
(46, 250),(48, 305),(50, 350),(52, 390),(54, 410)
39. Use linear regression to determine a function y, where the profit in thousands of dollars depends on the number of units sold in hundreds.
40. Predict when the profit will exceed one million dollars.
For the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The following ordered pairs show dollars and the number of units sold in hundreds and the profit in thousands of over the ten-year span (number of units sold, profit) for specific recorded years:
(46, 250), (48, 225), (50, 205), (52, 180), (54, 165).
41. Use linear regression to determine a function y, where the profit in thousands of dollars depends on the number of units sold in hundreds.
42. Predict when the profit will dip below the $25,000 threshold.
43. Determine whether the algebraic equation is linear. [latex]2x+3y=7\\[/latex]
44. Determine whether the algebraic equation is linear. [latex]6{x}^{2}-y=5\\[/latex]
45. Determine whether the function is increasing or decreasing.
[latex]f\left(x\right)=7x - 2\\[/latex]
46. Determine whether the function is increasing or decreasing.
[latex]g\left(x\right)=-x+2\\[/latex]
47. Given each set of information, find a linear equation that satisfies the given conditions, if possible.
Passes through [latex]\left(\text{7},\text{5}\right)\\[/latex] and [latex]\left(\text{3},\text{17}\right)\\[/latex]
48. Given each set of information, find a linear equation that satisfies the given conditions, if possible.
x-intercept at [latex]\left(\text{6},0\right)\\[/latex] and y-intercept at [latex]\left(0,\text{1}0\right)\\[/latex]
49. Find the slope of the line shown in the line graph.
50. Find the slope of the line graphed.
51. Write an equation in slope-intercept form for the line shown.
52. Does the following table represent a linear function? If so, find the linear equation that models the data.
| x |
–4 |
0 |
2 |
10 |
| g(x) |
18 |
–2 |
–12 |
–52 |
53. Does the following table represent a linear function? If so, find the linear equation that models the data.
| x |
6 |
8 |
12 |
26 |
| g(x) |
–8 |
–12 |
–18 |
–46 |
54. On June 1st, a company has $4,000,000 profit. If the company then loses 150,000 dollars per day thereafter in the month of June, what is the company’s profit nth day after June 1st?
For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:
55. [latex]\begin{cases}2x - 6y=12\hfill \\ -x+3y=1\hfill \end{cases}\\[/latex]
56. [latex]\begin{cases} y=\frac{1}{3}x - 2\hfill \\ 3x+y=-9\hfill \end{cases}\\[/latex]
For the following exercises, find the x- and y- intercepts of the given equation
57. [latex]7x+9y=-63\\[/latex]
58. [latex]f\left(x\right)=2x - 1\\[/latex]
For the following exercises, use the descriptions of the pairs of lines to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?
59. Line 1: Passes through [latex]\left(5,11\right)\\[/latex] and [latex]\left(10,1\right)\\[/latex]
Line 2: Passes through [latex]\left(-1,3\right)\\[/latex] and [latex]\left(-5,11\right)\\[/latex]
60. Line 1: Passes through [latex]\left(8,-10\right)\\[/latex] and [latex]\left(0,-26\right)\\[/latex]
Line 2: Passes through [latex]\left(2,5\right)\\[/latex] and [latex]\left(4,4\right)\\[/latex]
61. Write an equation for a line perpendicular to [latex]f\left(x\right)=5x - 1\\[/latex] and passing through the point (5, 20).
62. Find the equation of a line with a y- intercept of [latex]\left(0,\text{ }2\right)\\[/latex] and slope [latex]-\frac{1}{2}\\[/latex].
63. Sketch a graph of the linear function [latex]f\left(t\right)=2t - 5\\[/latex].
64. Find the point of intersection for the 2 linear functions: [latex]\begin{cases}x=y+6\\ 2x-y=13\end{cases}\\[/latex]
65. A car rental company offers two plans for renting a car.
Plan A: 25 dollars per day and 10 cents per mile
Plan B: 50 dollars per day with free unlimited mileage
How many miles would you need to drive for plan B to save you money?
66. Find the area of a triangle bounded by the y axis, the line [latex]f\left(x\right)=10 - 2x\\[/latex] , and the line perpendicular to f that passes through the origin.
67. A town’s population increases at a constant rate. In 2010 the population was 55,000. By 2012 the population had increased to 76,000. If this trend continues, predict the population in 2016.
68. The number of people afflicted with the common cold in the winter months dropped steadily by 50 each year since 2004 until 2010. In 2004, 875 people were inflicted. Find the linear function that models the number of people afflicted with the common cold C as a function of the year, t. When will no one be afflicted?
For the following exercises, use the graph in Figure 1 showing the profit, y, in thousands of dollars, of a company in a given year, x, where x represents years since 1980.
Figure 1
| Year |
Pima Central |
East Valley |
| 1970 |
32,000 |
120,250 |
| 2010 |
85,000 |
150,000 |
74. In which subdivision have home values increased at a higher rate?
75. If these trends were to continue, what would be the median home value in Pima Central in 2015?
76. Draw a scatter plot for the data in the table below. Then determine whether the data appears to be linearly related.
| 0 |
2 |
4 |
6 |
8 |
10 |
| –105 |
–50 |
1 |
55 |
105 |
160 |
77. Draw a scatter plot for the data in the table below. If we wanted to know when the population would reach 15,000, would the answer involve interpolation or extrapolation?
| Year |
Population |
| 1990 |
5,600 |
| 1995 |
5,950 |
| 2000 |
6,300 |
| 2005 |
6,600 |
| 2010 |
6,900 |
78. Eight students were asked to estimate their score on a 10-point quiz. Their estimated and actual scores are given in the table below. Plot the points, then sketch a line that fits the data.
| Predicted |
Actual |
| 6 |
6 |
| 7 |
7 |
| 7 |
8 |
| 8 |
8 |
| 7 |
9 |
| 9 |
10 |
| 10 |
10 |
| 10 |
9 |
79. Draw a best-fit line for the plotted data.
For the following exercises, consider the data in the table below, which shows the percent of unemployed in a city of people 25 years or older who are college graduates is given below, by year.
| Year |
Percent Graduates |
| 2000 |
6.5 |
| 2002 |
7.0 |
| 2005 |
7.4 |
| 2007 |
8.2 |
| 2010 |
9.0 |
80. Determine whether the trend appears to be linear. If so, and assuming the trend continues, find a linear regression model to predict the percent of unemployed in a given year to three decimal places.
81. In what year will the percentage exceed 12%?
82. Based on the set of data given below, calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to three decimal places.
| x |
17 |
20 |
23 |
26 |
29 |
| y |
15 |
25 |
31 |
37 |
40 |
83. Based on the set of data given below, calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to three decimal places.
| x |
10 |
12 |
15 |
18 |
20 |
| y |
36 |
34 |
30 |
28 |
22 |
For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs show the population and the year over the ten-year span (population, year) for specific recorded years:
(3,600, 2000); (4,000, 2001); (4,700, 2003); (6,000, 2006)
84. Use linear regression to determine a function y, where the year depends on the population, to three decimal places of accuracy.
85. Predict when the population will hit 12,000.
86. What is the correlation coefficient for this model to three decimal places of accuracy?
87. According to the model, what is the population in 2014?Licenses & Attributions
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- Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..
