Question

A hotel has 8 air conditioners that each have a pre- determined area liberation rate. The carbon dioxide (COx) emission rates (in ppm) are measured for each. The pollution rate is expected to be a linear function of the area liberation rate.

(a) Write out the equation of the regression line. Interpret the slope and intercept in the context of this problem. Do they make sense? Include a scatter plot of the data with the correct regression line added.

(b) Test the hypothesis that the linear relationships exist between the predictor and response variable (ANOVA, t-test for β1, t-test for ρ, or a confidence interval for β1).

(c) What is the R2 for the SLR you have obtained? What does the value mean? Use it to evaluate the linear model.

(d) Plot the standardized residuals against the independent variable. What can you say about the regression using this graph? (HINT: Are there outliers? Does it seem reasonable to claim the data has a linear fit?)

Area Liberation Rate	Carbon Dioxide Emission Rate
100	131
100	133
125	169
125	178
150	207
150	203
175	256
175	257
200	306
200	298
225	341
225	350
250	399
250	387
275	437
275	426
300	483
300	478
350	565
350	564
400	654
400	655
450	737
450	745

Answer 1

> area_liberation_rate=scan("clipboard")
Read 24 items
> carbon_dioxide_emmision_rate=scan("clipboard")
Read 24 items
> model=lm(carbon_dioxide_emmision_rate~area_liberation_rate)
> summary(model)

Call: lm(formula = carbon_dioxide_emmision_rate ~ area_liberation_rate) Residuals: Min 1Q Median 3Q Max -11.3546 -3.2579 -0.1612 3.8683 9.0417 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -49.05110 2.77576 -17.67 1.74e-14 *** area_liberation_rate 1.75604 0.01024 171.56 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 5.269 on 22 degrees of freedom Multiple R-squared: 0.9993,   Adjusted R-squared: 0.9992 F-statistic: 2.943e+04 on 1 and 22 DF, p-value: < 2.2e-16

> residuals=model$residuals
> plot(area_liberation_rate, residuals)

INTERPRETATIONS:

carbon_dioxide_emmision_rate = -49.0511 + 1.7560 * area_liberation_rate.

Here if the expected carbon_dioxide_emmision_rate is -49.0511 when area_liberation_rate is zero. The carbon_dioxide_emmision_rate is change in one unit change in carbon_dioxide_emmision_rate by 1.7560.

From ANOVA,

here p-value of F-statistic is 2.2e-16<0.05, we reject the null hypothesis, and conclude that the There is overall significant relationship between carbon_dioxide_emmision_rate and area_liberation_rate is exist.

From Multiple R-squared: 0.9993, here 99.93% variation in carbon_dioxide_emmision_rate is explain by the indpendent variable area_liberation_rate, it also indicate that the model is good.

From the graph of residual versus area_liberation_rate there no any pattarn exist, we conclude that there is homogeous variance assumption of regression analysis is satisf. and the given relationship is linear.