SAJC Check your Understanding for Linear Regression Teachers

Check your Understanding (Correlation and Linear Regression) MUTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1. In regression analysis, the variable that is being predicted is A the independent variable B the dependent variable C usually denoted by x D none of the above (B) 2. In the regression equation y = bo + b1x, bois the A slope of the line B independent variable C y intercept D none of the above (C) 3. In the regression equation y = bo + b1x, b1 is A the slope of the line B an independent variable C the y intercept D none of the above (A) 4. In regression analysis, the variable that is doing the predicting or explaining is A the independent variable B usually denoted by y C the dependent variable D none of the above (A)

5. The range of the correlation coefficient is A 0 to +1 B -1 to 0 C -1 to 0 D -1 to +1 (D) 6. If the slope of the regression equation y = bo + b1xis positive, then A as x increases y decreases B as x increases so does y C Either a or b is correct D none of the above (B) Scatter Diagram and Product Moment Correlation Coefficient 1. MI H1 Prelim 2017/Q11 An electric heater was switched on in a cold room and the temperature of the room was noted at five-minute intervals. Time from switching on electric heater, x (min) 0 5 10 15 20 25 30 35 40 Temperature of room, y (oC) 0.4 1.5 3.4 5.5 7.7 9.7 11. 7 13. 5 15. 4 (i) Draw a sketch of the scatter diagram for the data, as shown on your calculator. [2] (ii) Find the product moment correlation coefficient and comment on its value in the context of the data. [2] Answers (ii) 0.999 Solution:

1 (i) (ii) From graphing calculator, required coefficient, r ≈ 0.99870 = 0.999. (3 s.f) Since r > 0 and |r| = 0.999 is very close to 1, the value of r suggests that there is a strong positive linear correlation between the temperature of the room and the time from switching on the electric heater. 2. SAJC H1 Prelim 2017/Q7 (a) Eight pairs of values of variables x and y are measured. Draw a sketch of a possible scatter diagram of the data for each of the following cases: (i) the product moment correlation coefficient is approximately − 0.9, [1] (ii) the product moment correlation coefficient is approximately zero. [1] (b) A researcher recorded the water temperature T, in oC, and the depth D, in metres, at noon on a certain day at each of the eight locations in a lake. The results are summarized in the table below. D (m) 10 50 80 120 200 250 340 400 T (oC) 25.0 23.0 22.2 k 16.4 12.4 10.0 4.0 (i) It is known that the regression line of T on D is given by 0.051424 25.908.TD=− + Show that the value of k is 19.7, correct to 1 decimal place. [1] (ii) Give a sketch of the scatter diagram for the data. [2] (iii) Calculate the product moment correlation coefficient for the revised data and comment on its value in the context of the question. [2] (iv) Sketch the regression line T on D on your scatter diagram. [1] O y (oC) x (min) (40, 15.4) 5 40 15.4 1.5

Answers (iii) 0.992r=− (iv) 0.0514 25.9TD=− + Solution 2 ( ) Using T 0.051424 25.908, 1450 113substitute 181.25, 88 113 0.051424 181.25 25.9088 19.6992 19.7 (shown) D kDT k k =− + += = = + =− +  = = r  0.9 y x         y x         y         x x y         r  0

0.992r=− . It indicates a strong negative linear correlation between the depth of water and the temperature of water. As the depth of water increases, the temperature of water decreases. 3. TJC H1 Prelim 2017/Q8 The number of hours, x, spent daily on revision for mathematics and the marks, y, obtained for the mathematics year-end examination are recorded for 10 randomly selected students. The results are given in the following table. x 1.3 2.1 1.1 2.3 2.7 1.2 3.2 3.4 3.0 2.5 y 68 74 64 76 75 66 85 81 86 75 (i) Give a sketch of the scatter diagram for the data, as shown on your calculator. [2] (ii) Find the product moment correlation coefficient and comment on its value in the context of the data. [2] Answers (i) (ii) 0.938r= 68.5y= T D 5 10 15 20 25 30 35 40 5 1 1 2 0 2 3 (10,25 (400,4

Solution 3 (i) (ii) 0.938r= This indicates a strong positive linear correlation between revision hours and the marks obtained, i.e. as the hours for revision increases, the marks obtained increases linearly. 4. TPJC H1 Prelim 2017/Q9 The year x, and the mean maximum air temperature y, in degrees Celsius, of Singapore, are given in the following table. x 1974 1976 1980 1984 1989 1992 1998 2002 y 30.3 30.7 31.0 30.8 31.2 31.5 32.1 32.0 (i) Give a sketch of the scatter diagram for the data, as shown on your calculator. [2] (ii) Find the product moment correlation coefficient and comment on its value in the context of the data. [2] Answers (i) (ii) 0.960r=

Since r is close to 1, there is a strong positive linear relationship between the year and the mean maximum air temperature. In particular, as the year increases, the mean maximum air temperature increases. Solution 4 (i) 4 (ii) Using GC, 0.960r= Since r is close to 1, there is a strong positive linear relationship between the year and the mean maximum air temperature. In particular, as the year increases, the mean max imum air temperature increases. 5. TPJC H1 Prelim 2017/Q9 The Mathematics score, x, and the English score, y, of 8 Primary Four students during a year end examination are given in the following table. Studen t A B C D E F G H x 37 41 49 52 53 57 72 75 y 73 64 53 65 50 57 65 45 (i) Give a sketch of the scatter diagram for the data, as shown on your calculator. [2] (ii) Find the product moment correlation coefficient. [1] (iii) The least squares regression line of y on x is used to calculate an estimate of the English score for a student who scored 48 for Mathematics. State, with reasons, whether the estimate will be a reliable one. [2] (1974, 30.3) (2002, 32) y x

Solution 5)(i) (ii) r = – 0.53382 (iii) The estimate is unreliable as from the scatter diagram, the points do not seem to lie close to straight line and r is not close to – 1. Answers (i) (ii) r= – 0.53382 (iii) unreliable y x

Use the appropriate least squares regression line to predict or estimate unknown values 1. MI H1 Prelim 2017/Q11 An electric heater was switched on in a col