RI 2025 H2 S6 Corr and Reg Lecture Notes

RAFFLES INSTITUTION H2 Mathematics 9758 Year 6 2025 ________________________________ Chapter S6: Correlation and Regression Page 1 of 26 Chapter S6: Correlation and Regression Syllabus will include • Use of scatter diagram to judge if there is a plausible linear relationship between the two variables • Correlation coefficient as a measure of the fit of a linear model to the scatter diagram • Interpreting the product moment correlation coefficient (in particular, values close to −1, 0 and 1) • Concepts of linear regression and method of least squares to find the equation of the regression line • Concepts of interpolation and extrapolation • Use of the appropriate regression line to make prediction or estimate a value in practical situations, including explaining how well the situation is modelled by the linear regression model • Use of a square, reciprocal or logarithmic transformation to achieve linearity CONTENT 1 Terminology 1.1 Describing Variables 1.2 Scatter Diagram 1.2.1 Producing a Scatter Diagram using the Graphing calculator 1.2.2 Interpreting Scatter Diagrams 2 Product Moment Correlation Coefficient 2.1 Calculating the Value of the Product Moment Correlation Coefficient 2.2 Properties of the Product Moment Correlation Coefficient 2.3 Interpreting Correlation 2.3.1 Importance of Scatter Diagram 2.3.2 Correlation does not imply Causation 3 Linear Regression 3.1 Least Squares Regression Line of y on x 3.2 Least Squares Regression Line of x on y 3.3 Calculating the Least Squares Regression Lines and Product Moment Correlation Coefficient using the Graphing calculator 4 Application and Interpretation 5 Linearisation of Data Appendix: Equivalence of the 2 Formulae for the Product Moment Correlation Coefficient

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ________________________________ Chapter S6: Correlation and Regression Page 2 of 26 INTRODUCTION Suppose that you are helping a proud mother to keep track of the weight of her newborn. After monitoring the infant’s weight ( y kg) for a period of time ( t days), you obtain the following data: t 7 32 71 97 188 273 409 y 4.43 4.88 6.31 7.18 10.63 13.60 17.95 Can you then tell the mother how much her baby will weigh by the next month? What about the next year? How certain would you be of your prediction? Can you further use your data to check if the infant’s growth is normal? In this chapter, we will deal with the skills required to answer the above questions. Statistical inference entails the study of correlation between variables and regression analysis, the art and science of finding these relationships between variables. Empirical data rarely corroborate exactly with theoretical mathematical models, hence the need for statistical analysis to accommodate such deviations. 1 TERMINOLOGY 1.1 Describing Variables • The data in the above example are a set of pairs of values for two variables. This is an example of bivariate data, where each observation requires the values of two variables. • Since an infant’s weight is dependent upon his/her age, t is the independent variable and y is the dependent variable in the above example. • Sometimes the independent variable is controlled so that the variable only assumes a set of predetermined values. Example 1 Identify the dependent and independent variables in each of the scenarios below. (a) An object is heated up and its volume measured at different temperatures. The following table shows eight of these measurements. Temperature (C), t 30 40 50 60 70 80 90 100 Volume (m3), v 1.000 1.256 1.395 1.523 1.648 1.699 1.712 1.735 Ans: The independent variable is t , and the dependent variable is v . Note: A controlled variable is an independent variable.

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ________________________________ Chapter S6: Correlation and Regression Page 3 of 26 (b) At a restaurant, the waiting time is defined as the time between sitting down at a table and a waiter first arriving at your table. This waiting time is dependent upon the number of other customers already seated in the restaurant. John is a customer who visited the restaurant on 10 different days. The table shows, for each of these days, the number, ,x of customers already seated and his waiting time, in y minutes. x 9 3 4 10 8 12 7 11 2 6 y 11 6 5 11 9 13 9 12 4 7 (Taken from GCE Jan 2006 ASE) Ans: The independent variable is x , and the dependent variable is y . (c) The following table shows the marks scored by 10 randomly selected students in a French test and in an English test. French score, x 20 43 33 56 50 67 73 68 77 43 English score, y 19 42 44 52 51 53 66 56 60 37 Ans: From the above information, we are not able to determine which is dependent/independent variable. 1.2 Scatter Diagram A picture speaks a thousand words – what better way to obtain an overview of the relationship between variables than observing it from a diagram. A scatter diagram is a two -dimensional plot, with the values of one variable plotted along each axis. By convention, we plot the independent variable along the horizontal axis.

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ________________________________ Chapter S6: Correlation and Regression Page 4 of 26 1.2.1 Producing a Scatter Diagram using the Graphing calculator Consider the following set of data, with x as the independent variable and y as the dependent variable. x 9 3 4 10 8 12 7 11 2 6 y 11 6 5 11 9 13 9 12 4 7 Using the Graphing Calculator to produce a scatter diagram: 1. Press …Í to go to the List editor. 2. Key the data into L1 and L2. 3. Press y, to turn on the Stat Plot function. 4. Xlist and Ylist refer to the list of variables on the horizontal and vertical axes respectively. In this example, ensure that the axes are set to correspond to the correct lists L1 and L2. 5. Press q® to view the scatter diagram with all statistical data points displayed. 6. Press r and the left and right arrows to see the coordinates of each data point.

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ________________________________ Chapter S6: Correlation and Regression Page 5 of 26 Note: When drawing a scatter diagram, ensure the relative position of the points are correct, and label the axes and range of the data: 1.2.2 Interpreting Scatter Diagrams If all the points in a scatter diagram seem to lie close to a straight line, we say there is a linear correlation between the variables. Both the variables increase together. The points lie close to a straight line. We say that the variables have a strong positive linear correlation. As one variable increase, the other decreases. There is a negative linear correlation between the variables. The points are further away from a straight line as compared to the first case. So the linear correlation here is not as strong as the first case. There appears to be no clear relation between the variables. There is a non-linear relation between the variables. y 13 4 2 12 x

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ________________________________ Chapter S6: Correlation and Regression Page 6 of 26 2 PRODUCT MOMENT CORRELATION COEFFI