The Correlation Coefficient and The Regression Equation

12.3Suppose of fat you want to know if there is a relationship between gramsconsumed consumed by college students. Also, if there is a relationship calories consumed calories andto estimate and grams of fat consumed by college students, between information be usedgrams of fat that are associated with a given amount could of this calories?What information-would-you-need?1, get dataData: Galories consumed A variableGrams of fale available0Examine 2.4 data (viswally analitealy to determineif there is a realship between the 2, variable A relation bad there not cause BeffedDivorce rate in Mainecorrelates withPer capita consumption of margarineCorrelation: 99 9.26% (r=0.992558)20002001200220032004200520062007200820094.95 per 1,000theMake4.62 per 1,000in6lbsThisDiverse4.29 per 1,0004 lby3.96 per 1,00023bs20002001200220032004200520062007200%2009Margarine consumedDivorce rate In MaloeData sources National Vital Statistics Reports 2nd U.S. Department of AgricultureTotal revenue generated by arcadesIIIcorrelates withComputer science doctorates awarded in the USCorrelation: 98.51% (r=0.985065)200020012002200320042005200620072008200952 billion2000 agrees$1.75 billion1500 degreesComputer$1.5 billionscience$1.25 billion1000 degrees$1 billion2000200120022003500 degrees200420052006200720082009Computer science doctoratesArcade revenueUnta sources U.S. Cansus Euranu and National Science FoundationGraphs from: Section 12.3 The Correlation Coefficient and The Regression EquationGoals:1) Know what correlation means (and what it doesn't)2) Use the Lin Reg Test function determine the correlation coefficient;3) Use the inReg I Test function determine the linear regression equation, y = a + bx4) Use a scatter plot to determine if there is linear correlation and the type (positive ornegative)Correlation: A relationship or an association between two or more variables.Correlation does NOT mean causation:cor-re-la-tionnouna mutual relationship or connection between two or more things: researchshowed a clear correlation between recession and levels of property crime.the process of establishing a relationship or connection between two or moremeasures: the increasingly similar basis underlying national soil maps allowscorrelation to take place more easily.Statistics interdependence of variable quantities.Statistics a quantity measuring the extent of the interdependence of variablequantities.cau-sa-tion kô'zäSH(a)nInounthe action of causing something: investigating the role of nitrate in the causationof cancer.the relationship between cause and effect; causality: a strong association isnot a proof of causation. Types of linear (and non-linear) correlation:addner(a) Positive linear pattern (strong)(b) Linear pattern wl one deviationCloseto the linearPostive abres cause theline is going up.Tread is thatthere droping/gethingbigger left to(a) Negative linear pattern (strong)(b) Negative linear pattern (weak)rightAll are very close.Most are not close.(a) Exponential growth pattern(b) No pallemStart small ZipcreaseskyrocktsNo care ctationbecausebecause nopattern. How do we measure correlation?XPYThe correlation coefficient, r:Numerical measure of the strength and direction of the association between bivariatedata, the independent (explanatory) variable X and dependent (response) variable y.Characteristics of r:1) F is always between 1 and 1, inclusive2) The closer r is to either -1 or 1, the stronger the linear relationship between X and y.co-linear3) If r = -1, there is a perfect negative relationship between X and y.4) If r = 1, there is a perfect positive relationship between X and y5) The closer r is to 0; the weaker the relationship is between X and y:The coefficient of determination, r2:What data points are close to linear thethe proportion (or percent) of variation in the dependent variable linear (y) thatcan Represents be explained by the variation in the independent variable (x) using theregression equation.1-r2LLR can not be explained by the variation in the independent variable (x) usingrepresents the proportion (or percent) of variation in the dependent variable the (y)that linear regression equation. (these are the points scattered around the line of best-fit)Characteristics of 21) i2is typically expressed as a percent:2) R is always between either 0.0 and 1, inclusive or 0% and 100%, inclusive. Computing r and the linear regression equation:Use LinRegTTest to generate:Inear regression equationExample.Regression equation, y = a + bx:NORMAL FLOAT AUTO REAL RADEAH CL- slope is b- y-intercept is (0,a).y=a+bxLAnRenTTest (F)Correlation coefficient, rB70 and PM0p-valuethe slope/ to -5. 103749139€ test Staffrate ofp=. 0363121268df=2changea=. 56152b=-.20768Y-inpect4s=.0909893071IINORMAL FLOAT AUTO REAL KADLAN CLLinRegTTesty=a+bxBF0 and PFO1df=2a=, 56152F Y. Inpectb=-. 20768 F Sbpcs=. 0909893071r2= 92869431792.9% A-9636878732\Very strong Stat 7 Edit1. Enter X for LlComputing [ and the linear regression equationkg t test1) plot Plot the points.Construct a scatter- Enter Y forlz2) Determine r (round to 3 decimal places)Window2. Set candow3) Determine r2 (round to the nearest tenth of a percent)4) Determine the linear regression model2nd5) Graph the linear regression model on the scatter plotturn in on4 G2Y= again make sureitsXy--65 (6,5,+> Y-mak=53 (-5,3)e-22 (-2,2)2 Ysc 1-1-10 (1/0)23 (2,-3)3-4 (3,44 J1x SCL1QX0x-maxX - min-10TY-min3. GraphCorrelation coefficient, r (round to 3 decimal places)2nd & Stals stestr2--0,9697 7 96.97% V Linerg testCloudr- -. 98597%Very strong / left to right points are-part dropLinear regression equation, y= a + bx (round a and b to two decimal places)y= 0.93+ - -0.95xxxline of best fit Computing r and the linear regression equation1) Construct a scatter plot2) Determine r (round to 3 decimal places)3) Determine r2 (round to the nearest tenth of a percent)4) Determine the linear regression model5) Graph the linear regression model. on the scatter plotr= 8868 7886878870Caloriesfat (g)205076184856r2=.942 vlavVer2335102Ysol=10212485Ymin-4056xmin=15,00Y - - 93,23+ 008x207431523462005772135892780146AInput into years cal00002150260026502700 2750