15.7 Curve Fitting

With statistical applications, exact relationships may not exist  Often an average relationship is used Regression Analysis: a collection of methods by which estimates are made between 2 variables Correlation Analysis: tells the degree to which the variables are related Scatter plot: good tool for recognizing & analyzing relationships dependent variable (prediction to be made) independent variable (the basis for the prediction)

Ex 1) A college admissions committee wishes to predict students’ first-year math averages from their math SAT scores. The data was plotted: The points are NOT on a clear straight line. But what can this plot tell us? - In general, higher SAT scores correspond to higher grades - Looks like a positive linear (or direct) relationship * Our graphing calculator can plot scatter plots Remember to identify which variable is independent & which is dependent. * Once we have a scatter plot, we identify the line or curve that “best fits” the points

Some examples:

Statisticians use the method of least squares to obtain a linear regression equation y = a + bx *the sum of all the values y is zero *the sum of the squares of the values y is as small as possible *On Calculator  2nd 0 (CATALOG)  Scroll down to Diagnostic On  Enter, Enter

Ex 2) A runner’s stride rate is related to his or her speed. Speed (ft/s) 15.86 16.88 17.50 18.62 19.97 21.06 22.11 Stride rate (steps/s) 3.05 3.12 3.17 3.25 3.36 3.46 3.55 Plot using a scatter plot STAT  1: Edit… L1 = enter data for speed (independent) L2 = enter data for stride rate (dependent) 2nd Y= (STAT PLOT)  1: Enter WINDOW GRAPH  looks like a line!

Do a “linear regression” STAT  CALC  8: Lin Reg (a+bx) Xlist: L1 Ylist: L2 Calculate enter  y = 1.766 + .080x What is r?  correlation coefficient It tells us “how good” our regression model fits. The closer it is to 1 (for direct linear) or –1 (for inverse linear), the better our model fits the data.

Ex 3) Match the scatter diagrams with the correlation coefficients r = 0.3, r = 0.9, r = –0.4, r = –0.75 a) b) c) r = 0.9 rising line fits points well r = –0.4 falling line fits points but not closely r = 0.3 rising line fits points but not closely A straight line is not always the best way to describe a relationship. The relationship may be described as curvilinear. Exponential ExpReg y = abx Logarithmic LnReg y = a + b lnx Power PwrReg y = axb All of these can be found in the STAT  CALC menu

Scatter plot  STAT  1:EDIT  Enter data in L1 & L2 WINDOW GRAPH Ex 4) An accountant presents the data in the table about a company’s profits in thousands of dollars for 7 years after a management reorganization. Year Profits 1 150 2 210 3 348 4 490 5 660 6 872 7 1400 Scatter plot  STAT  1:EDIT  Enter data in L1 & L2 WINDOW GRAPH Looks exponential Let’s try an exponential regression STAT  CALC  0: ExpReg  y = 107.208 (1.439) x Now use this model to predict the profit for year 8. y = 107.208 (1.439)8 = 1971.1  $1,971,000