# Solved Examples on Linear Models

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(1.) Examine these scatter plots and determine whether each relation is linear or nonlinear.

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(b.)

(c.)

(d.)

(e.)

(f.)

(a.) The relation is nonlinear because the data points in the scatter plot does not follow the pattern of a straight line.
It follows the pattern of the curve of a quadratic model: concave down.

(b.) The relation is linear because the data points in the scatter plot follow the pattern of a straight line.
The straight line pattern is that of a negative slope.

(c.) The relation is linear because the data points in the scatter plot follow the pattern of a straight line.
The straight line pattern is that of a positive slope.

(d.) The relation is linear because the data points in the scatter plot follow the pattern of a straight line.
The straight line pattern is that of a negative slope.

(e.) The relation is nonlinear because the data points in the scatter plot does not follow the pattern of a straight line.
It follows the pattern of the curve of a quadratic model: concave down.

(f.) The relation is nonlinear because the data points in the scatter plot does not follow the pattern of a straight line.
It follows the pattern of the curve of a quadratic model: concave up.
(2.) For the data given below:

 x 3 4 5 6 7 8 9 y 3 5 6 9 11 13 15

(a.) Draw a scatter plot.
[Please see the use of Texas Instruments calculators to draw the scatter plot. Answer: (2.)(a.)]

(b.) Find the equation of the line containing the first and last data points.

(c.) Graph the line found in part​ (b.) on the scatter plot.
[Please see the use of Texas Instruments calculators to graph the line on the scatter plot. Answer: (2.)(c.)]

(d.) Use a graphing utility to find the line of best fit.
Type integers or decimals rounded to four decimal places as needed.
[Please see the use of Texas Instruments calculators to determine the regression equation (also known as the line of best fit). Answer: (2.)(d.)]

(e.) What is the correlation coefficient, r?
Type an integer or decimal rounded to three decimal places as needed.
[Please see the use of Texas Instruments calculators to determine the correlation coefficient. Answer: (2.)(e.)]

(f.) Use a graphing utility to draw the scatter plot and graph the line of best fit on it using these values:
Xmin = ​0, Xmax = ​20, Xscl = ​2, Ymin= ​0, Ymax = ​20, Yscl = 2

[Please see the use of Texas Instruments calculators to draw the scatter plot and the line of best fit. Answer: (2.)(f.)]

(a.) Please see the use of Texas Instruments calculators to draw the scatter plot. Answer: (2.)(a.)

$(b.) \\[3ex] First\;\;Point = (3, 3) \\[3ex] x_1 = 3 \\[3ex] y_1 = 3 \\[3ex] Last\;\;Point = (9, 15) \\[3ex] x_2 = 9 \\[3ex] y_2 = 15 \\[3ex] slope = m = \dfrac{y_2 - y_1}{x_2 - x_1} \\[5ex] = \dfrac{15 - 3}{9 - 3} \\[5ex] = \dfrac{12}{6} \\[5ex] = 2 \\[3ex] To\;\;find\;\;the\;\;y-intercept: \\[3ex] Use\;\;any\;\;point \\[3ex] Let\;\;us\;\;use\;\;Last\;\;Point = (9, 15) \\[3ex] x = 9 \\[3ex] y = 15 \\[3ex] y = mx + b \\[3ex] 15 = 2(9) + b \\[3ex] 15 = 18 + b \\[3ex] 18 + b = 15 \\[3ex] b = 15 - 18 \\[3ex] b = -3 \\[3ex] \implies \\[3ex] y = mx + b \\[3ex] y = 2x - 3 \\[3ex]$ (c.) Please see the use of Texas Instruments calculators to graph the line on the scatter plot: Answer: (2.)(c.)

(d.) Please see the use of Texas Instruments calculators to determine the line of best fit (also known as the least-squares regression line): Answer: (2.)(d.)

(e.) Please see the use of Texas Instruments calculators to determine the correlation coefficient: Answer: (2.)(e.)

(f.) Please see the use of Texas Instruments calculators to draw the scatter plot and graph the line of best fit using those settings: Answer: (2.)(f.)
(3.) Use a graphing utility to find the line of best fit for the following data.
Type integers or decimals rounded to four decimal places as​ needed.

 x 2 4 4 6 8 10 y 9 12 13 15 17 20

Please review the basic steps here: Texas Instruments (TI) Calculators for Linear Models

$y = ax + b \\[3ex] y = 1.307692308x + 6.923076923 \\[3ex] y \approx 1.3077x + 6.9231...rounded\;\;to\;\;4\;\;decimal\;\;places$
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