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15.7 Curve Fitting

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

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15.7 Curve Fitting

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  1. 15.7 Curve Fitting

  2. With statistical applications, exact relationships may not exist  Often an averagerelationship is used RegressionAnalysis: a collection of methods by which estimates are made between 2 variables CorrelationAnalysis: tells the degree to which the variables are related Scatterplot: good tool for recognizing & analyzing relationships dependent variable (prediction to be made) independent variable (the basis for the prediction)

  3. 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

  4. Some examples:

  5. Statisticians use the methodofleastsquares to obtain a linearregression 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

  6. Ex 2) A runner’s stride rate is related to his or her speed. 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!

  7. 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.

  8. Ex 3) Match the scatter diagrams with the correlation coefficients r = 0.3, r = 0.9, r = –0.4, r = –0.75 c) a) b) 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

  9. 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. Scatter plot  STAT  1:EDIT  Enter data in L1 & L2 WINDOW GRAPH Typo  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

  10. Homework #1507 Pg 842 #4 – 7, 9 – 12, 14, 17 – 26, 31, 38 (use your graphing calculator for all scatter plots then make a sketch of what your calculator gives)

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