1. Describing Bivariate Relationships

2. Bivariate Relationships When exploring/describing a bivariate (x,y) relationship: Determine the Explanatory and Response variables Plot the data in a scatterplot Note the Strength, Direction, and Form Note the mean and standard deviation of x and the mean and standard deviation of y Calculate and Interpret the Correlation, r Calculate and Interpret the Least Squares Regression Line in context. Assess the appropriateness of the LSRL by constructing a Residual Plot.

3. Corrosion and Strength Consider the following data from the article, “The Carbonation of Concrete Structures in the Tropical Environment of Singapore” (Magazine of Concrete Research (1996):293-300): x= carbonation depth in concrete (mm) y= strength of concrete (Mpa) x 820 303540505565 y 22.817.121.516.113.412.411.49.76.8 Define the Explanatory and Response Variables. Exp.________________Resp. _________________

4. Corrosion and Strength Depth (mm) Strength (Mpa) There is a strong, negative, linear relationship between depth of corrosion and concrete strength. As the depth increases, the strength decreases at a constant rate.

5. Corrosion and Strength Depth (mm) Strength (Mpa) The mean depth of corrosion is _____mm with a standard deviation of ______mm. The mean strength is _____ Mpa with a standard deviation of ______Mpa.

6. Least Squares Regression (LSRL) Regression Line – A straight line that describes how a response variable changes as an explanatory variable changes, used to predict values of y given x. Least Squares Regression - Method for finding a line that summarizes the relationship between two variables.

7. Least-Squares Regression Line (LSRL) Mathematical model for the data Must have or show a linear trend to use Sum of the areas of the squares of the vertical distances of the data points from the line needs to be as small as possible Error = ___________________________________ Equation: y hat is the predicted value, a is the y- intercept, and b is the slope.

8. LSRL Slope b = rate of range, r = correlation coefficient, s y = standard deviation of y values, and s x = standard deviation of x values. Y-intercept is found using the mean of the x and mean of the y values. Round the slope and y- intercept to 3 decimal places.

9. Facts about LSRL Must know which is the explanatory and which is the response variable or you could get the incorrect LSRL. A change of one standard deviation in x corresponds to a change of r standard deviations in y. If r=1 or r=-1, the change in the predicted value is the same in standard deviations as the change in x (sum of areas is very small). Otherwise, the change in the predicted is less than the change in x.

10. Corrosion and Strength Depth (mm) Strength (Mpa) y=_____ + _____x strength=_____+___depth r=________ What does “r” tell us? What does “b=-0.28” tell us?

11. Least Squares Method http://standards.nctm.org/document/eexamples/ch ap7/7.4/index.htm#applet http://standards.nctm.org/document/eexamples/ch ap7/7.4/index.htm#applet

12. How Residuals Fit In The vertical distance from each data point to the regression line is the error, or residual. Some points have positive residuals; some have negative ones. If all the points fell on the line, there would be no error and no residuals. The mean of the sample residuals is always 0 because the regression line is always drawn such that half of the error is above it and half below. The equations that you used to estimate the intercept and slope determine a line of “best fit” by minimizing the sum of the squared residuals. This method of regression is called least squares. Because regression estimates usually contain some error (that is, all points do not fall on the line), an error term ( ɛ, the Greek letter epsilon) is usually added to the end of the equation:

13. How do you know if using the LSRL is the best prediction for an x value? Sum of Squares about the Mean SSM Sum of Squares for Error SSE If SSM and SSE are about the same, then you need to use the mean of your y values as the best prediction. If SSE is very small then the LSRL is the best prediction.

14. Coefficient of Determination The difference SSM – SSE measures the amount of variation of y that can be explained by the regression line of y on x. This is the proportion of the total sample variability that is explained by the LSRL of y on x. It is also know as the coefficient of determination. Coefficient of Determination – fraction of the variation in the values of y that is explained by LSRL of y on x.

15. Corrosion and Strength Find the equation of the Least Squares Regression Line (LSRL) that models the relationship between corrosion and strength. Depth (mm) Strength (Mpa) y=24.52+(-0.28)x strength=24.52+(- 0.28)depth r=-0.96

16. Corrosion and Strength Use the prediction model (LSRL) to determine the following: What is the predicted strength of concrete with a corrosion depth of 25mm? strength=24.52+(-0.28)depth strength=24.52+(-0.28)(25) strength=17.59 Mpa What is the predicted strength of concrete with a corrosion depth of 40mm? strength=24.52+(-0.28)(40) strength=13.44 Mpa How does this prediction compare with the observed strength at a corrosion depth of 40mm?

17. Residuals Note, the predicted strength when corrosion=40mm is: predicted strength=13.44 Mpa The observed strength when corrosion=40mm is: observed strength=12.4mm The prediction did not match the observation. That is, there was an “error” or “residual” between our prediction and the actual observation. RESIDUAL = Observed y - Predicted y The residual when corrosion=40mm is: residual = 12.4 - 13.44 residual = -1.04

18. Assessing the Model Is the LSRL the most appropriate prediction model for strength? r suggests it will provide strong predictions...can we do better? To determine this, we need to study the residuals generated by the LSRL. Make a residual plot. Look for a pattern. If no pattern exists, the LSRL may be our best bet for predictions. If a pattern exists, a better prediction model may exist...

19. Residual Plot Construct a Residual Plot for the (depth,strength) LSRL. depth(mm) residuals There appears to be no pattern to the residual plot...therefore, the LSRL may be our best prediction model.

20. Coefficient of Determination We know what “r” tells us about the relationship between depth and strength....what about r 2 ? Depth (mm) Strength (Mpa) 93.75% of the variability in predicted strength can be explained by the LSRL on depth.

21. Summary When exploring a bivariate relationship: Make and interpret a scatterplot: Strength, Direction, Form Describe x and y: Mean and Standard Deviation in Context Find the Least Squares Regression Line. Write in context. Construct and Interpret a Residual Plot. Interpret r and r 2 in context. Use the LSRL to make predictions...