Lab 7 Linear and Nonlinear Regression
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Drexel University *
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Course
411
Subject
Mathematics
Date
Apr 3, 2024
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3
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Math 411 Lab 7: Linear and Nonlinear Regression 1 Overview In this lab, we’ll continue to examine linear and nonlinear regression techniques. We will model the relationship of a dependent variable to an independent variable. Complete all of the exercises for your lab report. As always, be sure to include all relevant information. For this lab, you should include your R code/scripts along with the output. 2 Regression Equations Exercises: 1.
The article ‘Characterization of Highway Runoff in the Austin, Texas Area’ provides the following data for ?
= rainfall volume (
𝑚
3
)
and ?
= runoff volume (
𝑚
3
) for a particular location.
?
?
6 8 12 12 15 15 17 17 22 18 30 28 39 30 48 50 54 41 66 50 73 57 82 73 96 75 113 93 126 102 a.
Does the use of a scatterplot of the data support the use of a simple linear regression model? b.
Calculate point estimates of the slope and intercept of the population regression line. c.
Calculate a point estimate of the true average runoff volume when rainfall volume is 50. d.
What proportion of the observed variation in runoff volume can be attributed to the simple linear regression relationship between runoff and rainfall? 2.
For the past decade, rubber powder has been used in asphalt cement to improve performance. The following table contains sample data where ?
= cube strength (MPa) versus ?
= axial strength (MPa).
?
?
112.3 75.2 97 70.2 93.7 56.9 85 46.9 101 73.5 100.2 72.5 95.8 68.2 95.5 58.5 88 55 85.7 46.7
a.
Obtain the equation of the least squares line, and interpret its slope. b.
Calculate and interpret the coefficient of determination. c.
Calculate and interpret an estimate of the error standard deviation 𝜎
in the simple linear regression model. 3.
Air pressure (psi) and temperature (
℉
) were measured for a compression process in a certain piston-cylinder device, resulting in the following data.
Pressure Temperature 15.0 37.9 37.0 89.4 57.8 134.3 81.2 152.8 103.4 182.2 125.3 209.4 141.1 218.4 161.4 240.5 186.9 260.4 204.4 260.8 224.8 278.5 243.8 280.3 264.1 301.3 275.4 318.3 302.1 311.8 322.6 333.0 340.1 328.6 358.8 335.9 a.
Would you fit the simple linear regression model to the data and use it as a basis for predicting temperature from pressure? Why or why not? b.
Find a suitable regression model and use it as a basis for predicting the value of temperature that would result from a pressure of 200, in the most informative way possible. c.
In words, state the null hypothesis and alternative hypothesis for the selected model. 4.
The accompanying data on ?
= frequency (MHz) and ?
= output power (W) for a certain laser configuration was read from a graph in the article ‘Frequency Dependence in RF Discharge
Excited Waveguide CO
2
Lasers’
.
?
?
59 14 63 16 78 17 101 19 126 20 157 17 186 14 223 4
a.
Does a quadratic model appear to be suitable for explaining observed variation in output power by relating it to frequency? b.
Would the simple linear regression model be nearly as satisfactory as the quadratic model? c.
Do you think it is worth considering a cubic model? d.
In words, state the null hypothesis and alternative hypothesis for the optimal model.
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