RI 2025 H2 S6 Corr and Reg Tutorial Qns

RAFFLES INSTITUTION H2 Mathematics 9758 2025 Year 6 _______________________________ Tutorial S6: Correlation and Regression Page 1 of 6 Tutorial S6: Correlation and Regression 1 The amounts, x grams, of catalyst used in a chemical reaction and the resulting times, y hours, taken to complete the reaction were recorded. The results are given in the following table. x 2.0 2.5 3.0 3.5 4.0 4.5 5.0 y 62.1 51.2 44.1 39.2 35.0 37.3 33.0 (i) Calculate the value of the product moment correlation coefficient for the data. (ii) State what your value indicates about the relation between x and y, and what you expect about the scatter diagram from this sample. (iii) Draw a scatter diagram of the data and comment on your answer to part (ii). [(i) 0.925 ] 2 The variables x and y are believed to be linearly related and a set of values of x and y are obtained experimentally. Both variables are subject to experimental error. State circumstances under which the regression line of x on y should be used, rather than the regression line of y on x, when the value of one variable is to be estimated from a given value of the other. The heart rate (x) and diastolic blood pressure (y), both in suitable units, were measured for each of 10 hospital patients after being given a certain drug. The results were as follows. x 49 51 54 58 63 64 68 70 75 78 y 90 88 85 91 82 85 76 77 70 71 (i) By referring to the scatter diagram of the data obtained using your graphing calculator, state what the scatter diagram indicates about the correlation between x and y. (ii) Calculate the equation of the estimated least square regression line of y on x in the form y a bx  , giving the values of a and b correct to 3 significant figures. (iii) Use a suitable regression line to estimate the heart rate when the diastolic blood pressure is 80. Give a reason why it might be unwise to use either of the regression lines to estimate the diastolic blood pressure when the heart rate is 90. (iv) Find the product moment correlation coefficient for the data and state, giving a reason, whether its value confirms your statement in part (i). [ (ii) 126 0.704y x  (iii) 64.8 (iv) 0.919 ]

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ ________________________________ Tutorial S6: Correlation and Regression Page 2 of 6 3 The following data is collected from a trial of a weedkiller. Trial field areas, each of area 1 hectare, were treated with different volumes of weedkiller. The volume of weedkiller applied is v litres, and x is the number of weeds found in the corresponding hectare after the weedkiller is applied. (i) Calculate the equation of the regression line of x on v in the form x a bv  . (ii) Use your answer to estimate the number of weeds that would be found in an area of 1 hectare treated with 35 litres of weedkiller. (iii) Interpret the coefficient b in the context of volume of weedkiller and the number of weeds. (iv) Give a reason why a may not necessarily give the expected number of weeds per hectare when no weedkiller is used. (v) Explain why in this context a linear model would probably not be appropriate for large values of v. [(i) 43.4 0.410x v  ; (ii)29] 4 With the aid of a suitable diagram, describe the difference between the regression line of Y on X and that of X on Y. [3] Under what circumstances would the above two lines be coincident? [1] The following summarizes the data from 10 sets of lengths(x) and breadths(y) in mm: 2 21782, 1483, 318086, 220257, 264582x y x y xy         (i) Find the value of the product moment correlation coefficient. [1] (ii) Find the equation of the regression line of y on x and that of x on y. [2] (iii) State a data point which can be included such that r remains the same for the new set of data. [1] (iv) Predict the breadth when the length is 185 mm. Comment on the reliability of your prediction. [2] [(i) 0.744; (ii) 0.584 44.3y x  , 0.949 37.4x y  ; (iii) 178.2,148.3; (iv) 152] 5 (a) Draw separate scatter diagrams, each with 8 points, all in the first quadrant, which represents the situation where the product moment correlation coefficent between variable x and y is (i) −1, (ii) 0, (iii) between 0.5 and 0.9. [3] v 10 20 30 40 50 60 70 80 90 x 46 32 31 27 18 15 14 12 11

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ ________________________________ Tutorial S6: Correlation and Regression Page 3 of 6 (b) An investigation into the effect of a fertiliser on yields of corn found that the amount of fertiliser applied, x, resulted in the average yields of corn, y, given below, where x and y are measured in suitable units. x 0 40 80 120 160 200 y 70 104 118 119 126 129 (i) Draw a scatter diagram for these values. State which of the following equations, where a and b are positive constants, provides the most accurate model of the relationship between x and y. (A) 2y ax b  (B) 2 ay bx  (C) ln 2y a x b  (D) y a x b  [2] (ii) Using the model you chose in part (i), write down the equation for the relationship between x and y, giving the numerical values of the coefficients. State the product moment correlation coefficient for this model. [3] (iii) Give two reasons why it would be reasonable to use your model to estimate the value of y when 189x . [2] 6 The radiation intensity I at time t , from a radioactive source, is given by the formula 0e ktI I  , where 0I and k are constants. Show that the relation between lnI and t is linear. The following data were obtained from a particular source. The values of t may be considered to be exact, while the values of I are subject to experimental error. t 0.2 0.4 0.6 0.8 1.0 I 3.22 1.63 0.89 0.41 0.36 (i) Find the equation of the estimated regression line of lnI on t and hence give estimates for 0I and k. (ii) Calculate the radiation intensity that would be expected at time 0.5t . (iii) Calculate the product moment correlation coefficient between t and lnI . (iv) It is required to estimate the value of t for which 1.5I . Explain why neither the regression equation of t on I nor the regression line of t on ln I should be used. (v) Use a regression line to give the best estimate that you can of the time when the radiation intensity is 1.5. [ ln 2.88 1.65I t   , 0 5.23I  , 2.88k , (ii) 1.24, (iii) 0.984 , (v) 0.433 ]

Raffles Institution H2 Mathematics 2025 Year 6 _____________________________________________________________________________________________ ________________________________ Tutorial S6: Correlation and Regression Page 4 of 6 7 Amy is revising for a mathematics examination and takes a different practice paper each week. Her marks, y% in week x, are as follows. (i) Draw a scatter diagram showing these marks. [1] (ii) Suggest a possible reason why one of the marks does not seem to follow the trend. [1] (iii) It is desired to predict Amy’s marks on future papers. Explain why, in this context, neither a linear nor a quadratic model is likely to be appropriate. [2] It is decided to fit a model of the form ln(L – y) = a + bx, where L is a suitable constant. The product moment correlation coefficient between x and ln(L – y) is denoted by r. The following table gives values of r for some possible values of L. L 91 92 93 r −0.929944 −0.929918 (iv) Calculate the value of r for L = 91, giving your answer correct to 6 decimal places. [1] (v) Use the table and your answer to part (iv) to suggest with a reason which of 91, 92 or 93 is the most appropriate