CHAPTER 7SIMPLE LINEAR REGRESSION1. INTRODUCTION2. THE REGRESSION EQUATION2.1. The Least Squares Regression Line2.2. The Predicted Value of y for a Given x Value, and the Residual e2.3. The Standard Error of Estimate2.4. Coefficient of Determination, R23. STATISTICAL INFERENCE FOR THE PARAMETERS OF POPULATION REGRESSION3.1. Confidence Interval for Population Slope Parameter β13.1.1. More About the Sampling Distribution of b13.2. Test of Hypothesis for Population Slope Parameter β13.3. Confidence Interval for Predicted Value of y for a Given x3.4. Confidence Interval for Mean Value of yfor a Given x4. USING EXCEL FOR REGRESSION ANALYSIS4.1. Understanding the Computer Regression Output1. INTRODUCTIONTo explain regression simply, suppose you want to find out what factor affects the students’ grades in the statistics departmental common finals. What determines the variations in test scores? Why do some studentshave higher scores than others? Suppose a friend offers the explanation that the variations in scores are related to students’ heights. Another friend proposes that score variations are related to the number of hours a student studies for the test. Your task is to find out which theory is more realistic (duh!).Here we have two models before us attempting to explain the variations in student statistics test scores. Each model consists of two variables: the dependent variable and the independent variable. In both models the dependent variable (also called the explained variable) is test scores. In Model 1, however, the independent variable (also called the explanatory variable) is student height and in Model 2 the independent variable is hours studied. Suppose you select a random sample of 10 students. You obtain the departmental final scores for the dependent variable. For the independent variable in Model 1 you measure their heights. For Model 2, you ask the students to state as accurately as possible the number of hours they studied for the test. The following are the hypothetical data for each model:Chapter 7—Regression Page 1 of 18Model 1 Model 2Variables VariablesDependent Independent Dependent IndependentyxyxHeight HoursScore in Inches Score Studied52 72 52 2.556 65 56 1.056 70 56 3.572 74 72 3.072 64 72 4.580 62 80 6.088 71 88 5.092 75 92 4.096 74 96 5.5100 69 100 7.0Your task is to find out which model better explains the variations in student scores. You do not observe the influence (if any) by looking at the numbers. In other words, it is hard to see any pattern or association in differences in scores in relation to differences in either height or hours studies. A visual aid is much more descriptive than the plain numbers. The visual aid in regression is called the scatter diagram. The following are the scatter diagrams for the two models. In each model the independent variable is measured on the horizontal axis and the dependent variable on the vertical axis.The scatter diagram for Model 1 shows that there is no relationship between student height and score, because there is no recognizable pattern. But in the scatter diagram for Model 2 there is a recognizable pattern showing that, in general, scores increase with the number of hours studied.Chapter 7—Regression Page 2 of 182. THE REGRESSION EQUATIONTo determine a more precise depiction of the relationship between the dependent variable y and the independent variable x, we need to describe the relationship as a mathematical equation. This equation is derived as the equation of the line that fits the scatter diagram the best. The regression analysis provides the tools for fitting a regression line onto the scatter diagram. To draw any line in the xy quadrant, you must have a vertical intercept and a slope. The general equation for a straight line is the following:y= b0+b1xHere b0 represents the vertical intercept and b1 the slope of the line. The slope represents the changein value of y per unit change in x:b1=∆ y∆ xOne can fit a line to the scatter diagram manually. There are many possible lines that could be fitted in this manner. However, there is a mathematical approach to fitting the most accurate, or best-fitting line. This method is called the least squares method. In explaining the method, you will see why it is called the least squares. We will use the data for Model 2 to explain how to find the regression equation.The model we are dealing with in this discussion is called a simple linear regression model. It is “simple“ because there is only one independent variable. If a model contains more than one independent variable thenit is called a multiple regression model. For example, in your model explaining the student scores, in addition to the number of hours studied, you may include the students’ SAT scores as a second independent variable. The current model is also a “linear“ regression model because the regression equation provides a straight regression line. The line is not curved. The general form of a simple linear regression equation is as follows:^y=b0+b1xNote that in the regression equation we use ^y (y-hat) rather than y. In regression models, the symbol y (hat-less) represents the observed values of the dependent variable, the actual value observed inthe sample. The distinction between y and ^y will become apparent below.Chapter 7—Regression Page 3 of 182.1. The Least Squares Regression LineThe mathematical method used to obtain the regression line is called the least squares method because with the resulting regression line the sum of squared value of the vertical distance between the observed y values andthe regression line is minimized (is the least). In the following diagram, the diamond-shaped markers represent the y values observed in the sample. For each value of x (hours) there is an observed value of y (actual score). The circular markers on the regression line represent the predicted values. Once you find the regression equation and draw the regression line, for each value of x there will be a corresponding predicted value of y on the line, which we denote by ^y.The vertical distance between the observed value ( y )and predicted value (^y ) is called the prediction error and is denoted by e: e= y−^y. Squaring the error terms and summing them we obtain the sum of squared errors. e2=( y −^y )2The least squares line assures that this sum of squares is minimized. You cannot find any other line that would provide a