36-402/608 Homework #6 due 10:30AM 2/251. Global warming (50 points)This problem uses the global warming data of Sleuth Chapter 15, case 2 fromcase1502.csv. See page 438 for a description.(a) Read in the data, and fit a simple regression model of temperature on year.Turn in the regression summary, and explain why the p-value for year mightbe incorrect.(b) Turn in a three-panel plot with panels for temperature vs. year, the ACF ofthe residuals, and the PACF of the residuals. For this and subsequent parts,be sure to override the main=, xlab=, and ylab= arguments to produce a plotsuitable for showing your boss. For example, nothing of the form “dtf$var”should show on the plots.(c) Fit three arima() models which include a linear trend in time and use thetime series models: ARMA(1,0), ARMA(2,0), and ARMA(0,1). Show yourcode and the three AIC values, and state which model is best according to theAIC criterion.(d) Using the best model from part c, turn in a plot with panels for the ACF ofthe residuals, the PACF of the residuals, and the cumulative periodogram ofthe residuals (don’t worry if the relative size of this plot looks funny). Use thefollowing code to place an appropriate “outer” title over the three sub-plots.par(mfrow=c(3,1), oma=c(0,0,2,0)) # save outer margin space at the top...mtext("My outer title", outer=TRUE, cex=1.4)(e) As an aid to learning about complex R objects, turn in the result of applyingnames() to your best arima() model object of part c. Also turn in the resultof myArimaObject$coef and myArimaObject$var.coef where “myArimaOb-ject” is whatever you called the model object. You may want to explore someother components of the object, as well as examining the “Value” portion of?arima to see what is available by using the “$” operator on an “arima” object.Note that not all of these are useful.(f) As an aid to learning about complex R objects, turn in the results of these twocommands, substituting your arima object for “myArimaObject”.class(myArimaObject)methods(class="Arima")Now you know that you can use, e.g., coef(myArimaObject), and vcov(myArimaObject)as an alternate way to get the coefficients and variance covariance matrix. Youcan also use tsdiag(myArimaObject) and predict(myArimaObject) to dothings not available with the “$” operator. Also note that the message “Non-visible functions are asterisked” indicates which method functions cannot bedirectly examined, even though they can be run and you can get help on someof them with “?”.(g) Turn in the plot that results from running tsdiag() on your best “arima”object of part c. Include an appropriate outer title. Note that finding all ofthe p-values > 0.05 in the Ljung-Box plot is a good indicator that you haveremoved all serial correlation.(h) Turn in the code to calculate the t-value and p-value and your statistical con-clusion for a test of no change in temperature over time vs. a change over timeusing code rather than directly entering numbers for the t ratio.(i) Turn in the t-value from directly entering the numbers that appear in theprintout of the arima object. Explain why it is different from the result of parth.(j) Explain what you might do to check if the temperature pattern over the 108years is curved rather than linear.2. Appliance Sales (50 points)(a) Load the appliance data from “appliances.dat” into a variable called “app”,throwing away or ignoring columns 6 and 7. Turn in an EDA plot of theUS sales (which are in thousands of appliances) over the available years fordishwashers, garbage disposals, refrigerators, and washing machines. Be sureto include 0 on the y-axis and to use different line types (lty=) so that theappliances can be distinguished in a black and white printout.(b) Which appliance most clearly requires advanced techniques due to a seasonal(yearly repeating) pattern?(c) We will examine dishwashers and disposals only. Will a linear trend be ade-quate?(d) Create columns in the data.frame for a centered version of year (i.e., year minusthe mean of year), and the square of that column.(e) Run an ordinary linear regression for each of the two appliances over timeincluding the square of time. Under what circumstances would the standarderror and p-value reported by lm() be correct vs. incorrect?(f) Turn in the ACF, PACF and cumulative periodogram for the residuals of thedishwasher model of part e.2(g) State your best guesses of the appropriate ARMA models for dishwashers anddisposals.(h) Fit the models of part g using arima() with xreg=cbind() binding togetherthe centered year variable and its square. Turn in the $coef components ofthese arima models for both outcomes.(i) Examine the ACF, PACF, and the cumulative periodogram of the arima resid-uals from part h for both outcomes, and turn in your conclusions.(j) Turn in the p-value for the test of a curved time course (vs. the null hypothesisof just a linear change over time) for dishwashers, along with your