Chapter 6

1. Chapter 6 Intercept A and Gradient of regression line, B

2. y – intercept or constant term Gradient or Slope of regression line y = A + Bx Dependent Variable, DV Independent Variable, IV y- axis y – intercept or constant term Gradient or Slope of regression line = B A x-axis

3. y = A + Bx is sometimes called a Deterministic model and it gives an exact relationship between y and x But in reality yobs is slightly different from the value predicted by ypre So y = A + Bx + e where e is random error term to take into consideration the difference (see slide 5 if you do not understand this concept) A and B are population parameters and the regression line is called Population regression line and values of A and B in the population are called true values of the y-intercept and slope. But population data are difficult to obtain. So we use sample data to estimate the population. Thus the values calculated from sample data are estimates and so the y-intercept and the slope for the sample data are denoted as ‘a’ and ‘b’ and yo is denoted as the predicted or estimated value for a given x. yo = a + bx this equation is called estimated regression model; it gives the regression of y on x based on sample data

4. Example 1 Income Food Expenditure 35 9 49 15 21 7 39 11 15 5 28 8 25 9

5. Scatter Plot for example 1 yobs Regression Line e ypre e (error) = ypre - yobs ypre – y value predicted by regression line or best straight line Yobs – actual y value obtained x1 Or e = y - yo

6. Error Sum of Squares, SSE The sum of errors is always zero for the best straight line or least squares line. i.e. Σe = Σ(y –yo) = 0 So to find the line that best fits the points, we cannot minimize the sum of errors Since it will always be zero. Instead we minimize the error sum of squares, SSE SSE = Σe2 = Σ(y –yo)2 The value of ‘a’ and ‘b’ that give the minimum SSE are called the least squares estimates of A and B and the regression line obtained with these estimates is called the least squares regression line. For the least squares regression line, yo = a + bx Where, b = SSxy and a = y - b x SSxx y = mean of y scores x = mean of x scores

7. SSxy = Σ (x - x)(y – y) SSxx = Σ (x – x)2 SSxx = Σx2 – (Σx)2 n Is always positive SSxy = Σxy – (Σx) (Σy) n Can be positive or negative y = mean of y scores x = mean of x scores

8. Example 1 Income Food x Expenditure, y xy x2 35 9 315 1225 49 15 735 2401 21 7 147 441 39 11 429 1521 15 5 75 225 28 8 224 784 25 9 225 625 Σx = 212 Σy = 64 Σxy = 2150 Σx2 = 7222 Step 1: Compute Σx, Σy, x and y. Σx = 212 Σy= 64 = Σx / n = 212 / 7 = 30.2857 = Σy / n = 64 / 7 = 9.1429 Step 2: Compute Σxy and Σx2

9. Step 3: Compute SSxy and SSxx SSxy = Σxy – (Σx) (Σy) n = 2150 – (212)(64) /7 = 211.7143 SSxx = Σx2 – (Σx)2 n = 7222 – (212)2 / 7 = 801.4286 Step 4: Compute ‘a’ and ‘b’ b = SSxy and a = y - b x SSxx a = 9.1429 – (.2642)(30.2857) = 211.7143 801.4286 a = 1.1414 = .2642 The estimated regression model ypre = a + bx is ypre = 1.1414 + .2642x

10. This gives the regression of food expenditure on income. Using this estimated regression model, we can find the predicted value Of y for any specific value of x. Eg. If the monthly income is RM3500, where x = 35 in hundred Then ypre = 1.1414 – (.2642)(35) = RM10.3884 hundred = RM1038.84 But the actual y value when x = 35 is RM900 There is an error in the prediction of –RM138.84 . This negative error indicates that the predicted value of y is greater than the actual value of y. Thus if We use the regression model, the household food expenditure is overestimated by RM138.84 Calculate what happens when income = RM0?

11. Exercise 1 Calculate the regression equation for the Math (x scores) and Science (y scores) marks.

12. Exercise 2 Calculate the regression equation for the Maths (x scores) and History (y scores) marks.

13. Exercise 3 Calculate the regression equation for the graph between IQ ranges (x axis) and the Correlation coefficients ( r) between Overall Creativity (OC) and Overall Achievement (OA) (y axis) based on the data on page 77 (Graph 7.1) (Palaniappan, 2006)