S CATTERPLOTS Correlation, Least-Squares Regression, Residuals Get That Program ANSCOMBE CRICKETS GESSEL

C ORRELATION Correlation is A measurement that combines the strength and direction of relationship Correlation Coefficient (r) The r value is a number that describes the strength & direction of the relationship You can find this value using your calculator. Diagnostics ON 2 nd 0 (catalog) x -1 Diagnostics On Enter Finding r Stat Calc 8: LinReg (a + bx)

R V ALUES StrongModerateWeak ModerateStrong Strong, Negative -1 = Perfect Negative Correlation Moderate, Negative Some, Negative Correlation Weak, Negative Not much Negative Correlation 0 = No Correlation Weak, Positive Moderate, Positive Strong, Positive Not much Positive Correlation Some, Positive Correlation 1 = Perfect Positive Correlation Remember, Correlation means Relationship Between the Variables

A RE THESE B ONES C ONNECTED ? Scientists are studying the relationship of the lengths of two different bones. Use the Correlation Coefficient to describe the relationship. FemurHumerus R =.9941 This means there is a strong positive relationship between the length of the femur and humerus bones.

F UN F ACTS ABOUT R r needs… The data have a linear pattern The data to be quantitative r doesn’t care about… Which variable is x and which is y Units of measurement r IS effected by outliers

Want to see the future? Then use the Least Squares Line… The Least Squares Line Straight Line (mathematical model) that best describes the scatter plot data of y based on x. Gives the smallest sum of the squared vertical distances from the points to the line. Least Squares Line is Your “Crystal Ball” By plugging in values into this line, you can predict values based on your collected data

L EAST S QUARES L INE Slopey - intercept Predicted value X = explanatory variable Y = response variable Stat: Calc: 8 Finds this equation on your calculator, but… We need to know how to do it by hand… Standard Deviation of y’s Standard Deviation of x’s Mean of y’sMean of x’s Use 2 Variable Stats to get the means and std. devs.

G RAPHING THE LSL ON YOUR S CATTER P LOT Using the Bone Data, Let’s look at the equation and then the graph of the scatter plot FemurHumerus Lin Reg Stats Then, go to “y=” VARS: 5(Statistics) Over to EQ, and Hit RegEQ: This puts your equation into your graph. Now just hit ZOOM 9.

H OW N OISY IS Y OUR C RICKET ? Scientists are investigating to see if the outside temperature has an effect on the rate of chirping by a cricket. Run the CRICKETS program to see the data collected… 2) Calculate by hand the Equation of the Least Squares Line 3) Use Your Calculator to Make a Scatter Plot of the Data and Plot the LSL on your graph. 1) Use Your Calculator to Calculate the Equation of the LSL. X-bar = 80.04; Y-bar = s x = 6.707; s y = 1.702; r =.835 =.2119 =-.3105

I NTERPRETING THE E QUATION IN C ONTEXT The Slope The rate of change The amount of change in y-hat when x increases by 1 The Intercept Value of y-hat when x = 0 Often the base value Back to The Crickets Describe the Slope and Y-Intercept in the context of the cricket problem. Slope: on average, each degree predicts.2119 more chirps Intercept: Not statistically significant because it would describe the number of chirps to be negative at 0 degrees.

H OW N OISY IS Y OUR C RICKET ? M AKING P REDICTIONS Using your equation, predict the rate of chirps per second for a cricket in: 90 degree weather 74 degree weather 60 degree weather chirps chirps chirps What About the Temperature if you hear 16.5 cricket chirps per second? 79 degrees

T HE C OEFFICIENT O F D ETERMINATION ( R 2 ) Tells the percent of the variation in the values of y that is explained by the equation Gives a measure of error in the equation Higher r 2 – More accurate predictions The r 2 value associated with our cricket example is.697. This says that 69.7% of the variation in cricket chirps is explained by the LS equation (temperature).

F ACTS A BOUT L EAST S QUARES L INE Must be clear on Explanatory & Response Variables Switching the variables changes your equation Line always passes through the point (x-bar,y- bar) This always gives us a point to start w/ or use during graphing Correlation is closely related to slope Smaller r = smaller effect of x on predictions r and r 2 help define the strength of a straight line relationship between the variables Higher values = stronger relationship

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