cPatrice Poage and Marcia Drost, February 1, 2008 1142, RegressionModeling Data: Which fits the best?Regression AnalysisA very useful skill is to be able to take data points and find a curve whichclosely approximates the given data. Finding that function is called regressionanalysis.Models to choose from: Enter the data in your calculator on L1and L2as illustrated in class,turn on the stat plot, and change the window to view the scatter plot. Then from the list below,find a curve which closely approximates the shape represented.• linear y = ax + b• quadratic y = ax2+ bx + c• cubic y = ax3+ bx2+ cx + d• quartic y = ax4+ bx3+ cx2+ dx + e• exponential y = abx, y > 0• power y = axb, x > 0, y > 0• logarithmic y = a + b ln(x), x > 0• logistic y =c1 + ae−bxNOTE: Exponential and logistic f unctions are never negative. Quadratic, quartic, and power func-tions can sometimes stay positive.How to choose the best model:• Graph the data points and decide which models do NOT make sense for the particular problem.• If the function is always increasing or always decreasing, compare the r values (to seewhich is closest to |1 |, a nd makes sense.• If the data increases and decreases, compare the r2values (to see which is closest to 1), as wellas, examine which models make more sense depending on the situation.• The best regression model, is a function that fits the data well, predicts the correct trend ofthe data outside your data set (in the future as well as the past), and is easy to solve.• If the data is used to find the best fit between points on the table, this is called interpolation.• If the data is used to find the best fit outside the given set of data, this is called extrapolation,and is often used to predict values in the future.Note: If the r or r2value does not show up on your screen, go to 2nd, 0, which is catalog, and scrolldown to diagnostics on, and hit enter twice.cPatrice Poage and Marcia Drost, February 1, 2008 2EX 1: Mr. Jones runs advertisements at a TV station to promote his used car sales. He collectsthe data relating advertising spots and the # of cars he sells per month as shown below:x (# of ads r un) 7 14 20 26 30 35 40y ( # of cars ) 14 20 25 28 29 29 30(a) If you were only concerned with the given data points, which of the following 3 models gives thebest fit and WHY: quadratic cubic logarithmic(b) If Mr. Jones wants to see long-term effects, which of the 3 models above makes the most sensethen? Explain WHY.EX 2: Given the following data for average cost of a new home in thousands of dollars, let x representthe year 1 960.Year 1 965 1968 1975 1980 1985 1990 1995Cost 21.5 26.6 40.1 75.5 11 0.2 145.6 176.2(a) Find the best fitting model, using quadratic, cubic, o r logistic, and explain why you think this isthe best model.(b) Using this model, predict the average cost in