Question

he following data is representative of that reported in an article on nitrogen emissions, with x - burner area liberation rate (MBtu/hr-ft2) and y NOx emission rate (ppm): x 100 125 125 150 150 200 200 250 250 300 300 350 400 400 у! 160 130 170 200 200 310 270 410 430 430 400 600 600 660 (a) Assuming that the simple linear regression model is valid, obtain the least squares estimate of the true regression line. (Round all numerical values to four decimal places.) (b) What is the estimate of expected NOx emission rate when burner area liberation rate equals 200? (Round your answer to two decimal places.) ppm (c) Estimate the amount by which you expect NOx emission rate to change when burner area liberation rate is decreased by 40. (Round your answer to two decimal places.) ppm (d) Would you use the estimated regression line to predict emission rate for a lberation rate of 500? Why or why not? O Yes, the data is perfectly linear, thus lending to accurate predictions. O Yes, this value is between two existing values. No, this value is too far away from the known values for useful extrapolation No, the data near this point deviates from the overall regression model.

Answer 1

Using R The commands for finding the least square regression line is

> x <- c(100,125,125,150,150,200,200,250,250,300,300,350,400,400)
> y <- c(160,130,170,200,200,310,270,410,430,430,400,600,600,660)
> relation <- lm(y~x)
> print(summary(relation))

Outout:

Coefficients:
Estimate Std. Error t value Pr(>|t|)   
(Intercept) -45.6833 25.9927 -1.758 0.104   
x 1.6999 0.1017 16.708 1.13e-09 ***

a) The regression line is ${\color{Blue} y=1.6999x-45.6833}$

b)The estimate of the NOx emission rate when $x=200$ is

$y\left ( 200 \right )=1.6999\left (200 \right )-45.6833\\ {\color{Blue} y\left ( 200 \right )=294.30 \textup{ ppm}}$

c) We have $\Delta y=1.6999\Delta x\\$ , When $\Delta x=-40$ ,

$\Delta y=1.6999\left ( -40 \right )\\ {\color{Blue} \Delta y=-68 \textup{ ppm}}$

d) Since the data is not perfectly linear, the data near $x=500$ deviates from the overall regression model.