1. Home
  2. Selected topics in algebra
  3. Linear functions
  4. Fitting linear models to data

2.3 Fitting linear models to data  (Page 5/14)

<< Chapter < Page
  Selected topics in algebra     Page 5 / 14
Chapter >> Page >

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.

y = a x + b a = 6.301 b = −1.044   r = −0.970

61.966 years

For the following exercises, draw a scatter plot for the data provided. Does the data appear to be linearly related?

0 2 4 6 8 10
–22 –19 –15 –11 –6 –2
1 2 3 4 5 6
46 50 59 75 100 136
Scatter plot with a collection of points appearing at (1,46); (2,50); (3,59); (4,75); (5, 100); and (6,136); they do not appear linear

No.

100 250 300 450 600 750
12 12.6 13.1 14 14.5 15.2
1 3 5 7 9 11
1 9 28 65 125 216
Scatterplot with a collection of points at (1,1); (3,9); (5,28); (7,65); (9,125); and (11,216); they do not appear linear

No.

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

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
Scatterplot with a collection of points at (16,46); (18,50); (20,54); (25,55); and (30,62); they appear nonlinear

Interpolation. About   60 ° F .

Graphical

For the following exercises, match each scatterplot with one of the four specified correlations in [link] and [link] .

Side-by-side scatter plots. The first is a scattered correlation in the positive direction. The second is a scattered correlation in the negative direction
Side-by-side scatter plots. The first has a strong negative correlation with all the points spaced out evenly near the top and center, but more spread out near the bottom. The second has a strong positive correlation, with the points more spread out near the bottom and closer together near the center and top.

r = 0. 95

r = −0. 89

This value of r indicates a strong negative correlation or slope, so C

r = −0.26

r = −0.39

This value of r indicates a weak negative correlation, so B

For the following exercises, draw a best-fit line for the plotted data.

Scatter plot with a domain of 0 to 10 and a range of 4 to 9. The points are at (0,5); (2.1,4.2); (3.5,6); (4.5,6.5); (5.5,6.8); (7,7.4); (8,8.5); (9,8); and (10,9).
Scatter plot with a domain of 0 to 10 and a range of -1 to 4. The points are at (0,1.5); (1.5, -0.1); (2.1,1.9); (3.4, 1.5); (4.5,2.5); (5.8,2.2); (6.8,3.8); (7.8,3.6); (8.8,2); and (10,2.4).
Scatter plot with domain 0 to 10 and a range from -1 to 4 with the line of best fit drawn going through the points: (0,1.5); (1.5, -0.1); (2.1,1.9); (3.4, 1.5); (4.5,2.5); (5.8,2.2); (6.8,3.8); (7.8,3.6); (8.8,2); and (10,2.4).
Scatter plot with a domain of 0 to 10 and range of 0 to 7 with the points: (0,7.3); (1,7); (2.2,6); (3.6,7); (4.8,6.2); (5.8,4); (6.6,3.8); (7.9,2.4); (8.8,2); and (10,0.1).
Scatter plot with a domain of 0 to 10 and a range of 2 to 6 with the points: (0,2.1); (1,3.9); (2.1,3.6); (3.6,3.9); (4.4,4); (5.6,4.2); (6.8,5); (7.8,5); (9,5.6); and (10,6).
Scatter plot with a domain of 0 to 10 and a range of 2 to 6 and the line of best fit going through the points: (0,2.1); (1,3.9); (2.1,3.6); (3.6,3.9); (4.4,4); (5.6,4.2); (6.8,5); (7.8,5); (9,5.6); and (10,6)

Numeric

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 [link] . Based on data from http://www.census.gov/hhes/socdemo/education/data/cps/historical/index.html. Accessed 5/1/2014. 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

The U.S. import of wine (in hectoliters) for several years is given in [link] . 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

Yes, trend appears linear because   r = 0. 985   and will exceed 12,000 near midyear, 2016, 24.6 years since 1992.

[link] 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

Technology

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.

x 8 15 26 31 56
y 23 41 53 72 103

y = 1 . 64 0 x + 13 . 8 00 , r = 0. 987

x 5 7 10 12 15
y 4 12 17 22 24
x y x y
3 21.9 10 18.54
4 22.22 11 15.76
5 22.74 12 13.68
6 22.26 13 14.1
7 20.78 14 14.02
8 17.6 15 11.94
9 16.52 16 12.76

y = − 0.962 x + 26.86 ,       r = − 0.965

<< Chapter < Page Page > Chapter >>
Practice Key Terms 5

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Selected topics in algebra. OpenStax CNX. Sep 02, 2015 Download for free at http://legacy.cnx.org/content/col11877/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Selected topics in algebra' conversation and receive update notifications?

Ask