SAJC 2022 Correlation and Linear Regression Lecture Notes (Teacher)

SAJC 2022 JC 2 H1 Mathematics Correlation and Linear Regression Page 1 of 59 Chapter 7 (Statistics) Correlation and Linear Regression Objectives At the end of the chapter, students should be able to: (a) Understand that bivariate data consists of the values of two variables (independent and dependent variables) obtained from the same sample expressed in ordered pairs; (b) Use the graphic calculator to plot the scatter diagram for a set of bivariate data to determine if there is a linear relationship between the two variables; (c) Understand that the cor relation coefficient is a measure of the fit of a scatter diagram to a linear model; (d) Calculate the product moment correlation coefficient for a set of bivariate data using a graphic calculator, and relate the value (in particular, values close to -1, 0 and 1) to the appearance of the scatter diagram; [Note: zero correlation does not necessarily imply ‘no relationship’, but rather ‘no linear relationship’.] (e) Understand that a high correlation between two variables does not necessarily imply one directly causes the other; (f) Understand the concepts of linear regression and ‘least squares’ with reference to the scatter diagram; (g) Use a graphic calculator to find the equation of the least squares regression line, and interpret its slope and intercept; [Note: A differ ent line will be obtained if we interchange the independent and dependent variables.] (h) Understand the concepts of extrapolation and interpolation of data, and use the appropriate regression line to make prediction or estimate a value in practical situations.

SAJC 2022 JC 2 H1 Mathematics Correlation and Linear Regression Page 2 of 59 Contents 7.1 Introduction 7.1.1 Bivariate Data 7.1.2 Independent vs Dependent Variables 7.2. Scatter Diagrams 7.2.1 Drawing Scatter Diagrams 7.2.2 Interpreting Scatter Diagrams 7.3 Product Moment Correlation Coefficient, r 7.3.1 What is Product Moment Correlation Coefficient, r? 7.3.2 Calculation of Product Moment Correlation Coefficient, r using Formula 7.3.3 Calculation of Product Moment Correlation Coefficient, r using Graphic Calculator 7.3.4 Properties of Product Moment Correlation Coefficient, r 7.3.5 Limitations of Product Moment Correlation Coefficient, r 7.3.6 The Effects of Transformation on Product Moment Correlation Coefficient, r 7.4 Linear Regression 7.4.1 Types of Regression Lines 7.4.2 Using Graphic Calculator to find Equations of Regression Lines 7.4.3 Least Squares Regression Lines of y on x 7.4.4 Least Squares Regression Lines of x on y 7.4.5 Properties of Regression Lines 7.5 Interpolation and Extrapolation 7.6 Self-Reading Examples Resources 1. A Concise Course in A Level Statistics with Worked Examples By J. Crawshaw and J. Chambers 2. A Comprehension Guide: H2 Mathematics for “A” Level, Volume 2 By Frederick Ho, David Khor, Yui-P’ng Lam and B.S. Ong 3. Worked Examples in Statistical Inference By R.N.D. Publications

SAJC 2022 JC 2 H1 Mathematics Correlation and Linear Regression Page 3 of 59 7.1 Introduction Regression analysis and correlation analysis have been developed to study and measure the statistical relationship that exists between two or more variables. In the A Level syllabus, we will investigate the linear regression analysis and linear correlation analysis of two variables. In linear correlation analysis, we measure the strength or closeness of the linear relationship between the two variables. In linear regression analysis , we prepare an estimating (or regression) linear e quation to estimate the values of one variable from given values of another. In other words, correlation analysis reveals the extent to which two variables are related whilst regression analysis tells us how they are related. EXAMPLE We are interested in the relationship between the number of hours studying and the marks obtained. The table below shows the amount of time ( x hours) that 6 average students spent studying for a test and the marks that they obtained (y out of 100 marks). x 1 2.5 3 4 4.2 5 y 20 46 48 60 61 65 Correlation and regression analyses will help us to answer the following questions: • Is there a relationship between the amount of time spent on revision and marks obtained? – Correlation • Is the relationship a linear one? – Linear Correlation • If there is a linear relationship, can we reasonably predict the value of one of the variables from the knowledge of the other? – By finding the regression line • If another student studies for 2 hours, can you predict how many marks he will obtain? What about if he studies for 7 hours? – By using the regression line • How certain are you of your prediction? – Interpolation, Extrapolation 7.1.1 Bivariate Data Bivariate data refers to data connecting two variables. Each observation thus comprises a pa ir of values taken by the two variables. It is customary to express the bivariate data as ordered pairs (x, y). An example of bivariate data is the data given in the table above, which shows a set of 6 pairs of values for the two variables, x and y.

SAJC 2022 JC 2 H1 Mathematics Correlation and Linear Regression Page 4 of 59 7.1.2 Independent vs Dependent Variables In the physical sciences, we often set up investigations or experiments in which we try to find a relationship between two variables. For such experiments, it is fairly common for the researcher to assign arbitrary values to one variable so that corresponding values of the other variable can be measured. For this reason, the variable that the researcher has control over is known as the independent variable (or controlled variable), while the other variable under inv estigation (whose values we want to predict) is called the dependent variable. On the other hand, there are situations (called observational studies) where the values of the independent variable cannot be pre -selected. An example of this is the study on t he effect of time taken to prepare for a test (independent variable) on result of the test (dependent variable). An independent variable (or input/controlled/predetermined variable) is usually the one whose value can be set or controlled. It is usually denoted by x. A dependent variable (or output/response variable) is usually the one whose value we want to predict. It is usually denoted by y. However, it is sometimes not possible to decide what the independent and dependent variables are, given a set of data. For example, History and Geography scores of a student. Example 1 Identifying independent and dependent variables In each of the cases below, identify the independent and dependent variables. (a) The drying time of a particular type of hobby paint depends on various factors including ambient temperature. The following table shows eight pairs of data obtained in an experiment on the ambient temperature in degree Celsius ( C ) and the corresponding drying time (in hours) of the paint. Ambient temperature, x ( C ) 32.6 6.6 23.5 12.7 35.1 20.3 27.8 17.4 Drying time, y (hours) 5.9 24.9 3.8 17.7 3.4 12.5 8.6 16.2 Independent variable : Ambient temperature Dependent variable : Drying time (b) The tabl e shows a Verbal Reasoning test score, x, and an English test score, y, for each of a random sample of 10 students who took both tests. Verbal Reasoning test score (x) 112 106 110 115 109 119 102 100 116 95 English test score (y) 69 63 75 79 72 85 90 58 75 60 Either x or y can be considered as the independent variables

SAJC 2022 JC 2 H1 Mathematics Correlation and Linear Regression Page 5 of 59 7.2 Scatter Diagrams 7.2.1 Drawing Scatter Diagrams A graph of the set of observations of the ordered pairs ( x, y) on