RI S6 Corr and Reg Add Prac Qns

RAFFLES INSTITUTION H2 Mathematics 9758 2025 Year 6 Additional Practice Questions for Chapter S6: Correlation and Regression 1 Seven cities in a certain country are linked by rail to the capital city. The table below shows the distance of each city from the capital and the rail fare from the city to the capital. City A B C D E F G Distance, x km 124 44 76 148 16 180 104 Rail fare, $y 156 53 99 169 23 177 138 (i) Give a sketch of the scatter diagram for the data as shown on your calculator. [2] (ii) Calculate the product moment correlation coefficient. [1] You are given that the regression line of y on x has equation 16.7 1.01y x  , where the coefficients are given correct to 3 significant figures. (iii) Calculate the equation of the regression line of x on y, giving your answer in the form x a by  . [1] (iv) Use the appropriate regression line to estimate (a) the rail fare from a city that is 28 km from the capital, [2] (b) the distance of a city from the capital if the rail fare is $198. [2] (v) Comment briefly on the reliability of the estimates in part (iv). [2] 2 An engineering company makes cranes. The numbers, x, sold in each three-month period for two years, together with the profits, y thousand dollars, on the sale of these cranes are given in the following table. (i) Give a sketch of the scatter diagram for the data as shown in your calculator. [2] (ii) Find x and ,y and mark the point  ,x y on your scatter diagram. [2] (iii) Calculate the equation of the regression line of y on x, and draw this line on your scatter diagram. [2] (iv) Calculate the product moment correlation, and comment on its value in relation to your scatter diagram. [2] (v) For the next three-month period, the sales target is 20 cranes. Estimate the corresponding profit. [2] (vi) The company’s sales director uses the regression line in (iii) to predict the profit if 40 cranes were to be sold in a three-month period. Comment on the validity of this prediction. [2] x 15 17 13 21 16 22 14 18 y 290 350 270 430 340 410 300 360

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ______________________________________________________ Additional Practice Questions for Chapter S6: Correlation and Regression Page 2 of 7 3 Explain why it is advisable to draw a scatter diagram before interpreting a correlation coefficient calculated for a sample drawn from a bivariate distribution. [1] Sketch a scatter diagram that shows an obvious relation between two variables but which yields a linear product-moment correlation coefficient that is close to zero. [1] 4 A medical officer wishes to investigate a patient’s heart-beat rate h beats per minute (bpm) when he walks at different speeds s km/h. The data is shown below: s (km/h) 1 1.5 2 2.5 3 3.5 4 4.5 5 h (bpm) 60 63 66 75 86 99 150 110 130 (i) Sketch a scatter plot of the above data. [1] (ii) One of the values of h appears to be incorrect. Indicate the corresponding point on your diagram by labelling it P. [1] Omit P for the remainder of this question. (iii) Calculate the product moment correlation coefficient for this set of data. Use the equation of an appropriate regression line to predict the value of s when h = 100, justifying your choice of regression line. [4] It is suggested to use one of the following two models instead: Model (I): h = a + bs2, Model (II): h = a + bes, where a and b are real constants. (iv) Determine which of the two models is a better choice, giving a reason for your answer. [2] (v) Suppose a new data pair ,s his added to the table above, where sand h are the patient’s sample mean walking speed (in km/h) and his sample mean heart-beat rate (in bpm) respectively, based on the data above. Without any calculations, explain whether the equation of the regression line you have obtained in part (iii) would change. [2]

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ______________________________________________________ Additional Practice Questions for Chapter S6: Correlation and Regression Page 3 of 7 5 (a) Sketch a scatter diagram that might be expected when x and y are related approximately by 2y px t  in each of the cases (i) and (ii) below. In each case your diagram should include 6 points, approximately equally spaced with respect to x, and with all x-values positive. (i) p and t are both positive. (ii) p is negative and t is positive. [2] (b) The ages in months (m) and prices in dollars (P) of a random sample of ten used cars of a certain model are given in the table. m 11 20 28 36 40 47 58 62 68 75 P 112800 102600 76500 72000 72000 69000 65800 57000 50600 47600 It is thought that the price after m months can be modelled by one of the formulae , ln ,P am b P c m d    where , , and a b c d are constants. (i) Find, correct to 4 decimal places, the value of the product moment correlation coefficient between (A) m and P, (B) ln m and P. [2] (ii) Explain which of and lnP am b P c m d    is the better model and find the equation of a suitable regression line for this model. [3] (iii) Use the equation of your regression line to estimate the price of a car that is 50 months old. [1] 6 Research is being carried out into how the concentration of a drug in the bloodstream varies with time, measured from when the drug is given. Observations of successive times give the data shown in the following table. Time (t minutes) 15 30 60 90 120 150 180 240 300 Concentration (x micrograms per litre) 82 65 43 37 22 19 12 6 2 It is given that the value of the product moment correlation coefficient for this data is 0.912 , correct to 3 decimal places. Draw a scatter diagram for the data. [1] Calculate the equation of the regression line of x on t. [2] Calculate the corresponding estimated value of x when 300t , and comment on the suitability of the linear model. [2] The variable y is defined by lny x . For the variables y and t , (i) calculate the product moment correlation coefficient and comment on its value, [2] (ii) calculate the equation of the appropriate regression line. [3] Use a regression line to give the best estimate that you can of the time when the drug concentration is 15 micrograms per litre. [2]

Raffles Institution H2 Mathematics 2025 Year 6 __________________________________________________________________________________________ ______________________________________________________ Additional Practice Questions for Chapter S6: Correlation and Regression Page 4 of 7 7 Mr. Lee recorded the length of time, y minutes, taken to travel to work when leaving home x minutes after 6.00 am on seven selected mornings. The results are as follows: x /min 0 10 20 30 40 50 60 y /min 16 27 28 39 43 48 51 (i) Calculate the equations of the least squares regression lines of y on x and x on y. [2] (ii) Mr. Lee forgot to add an observation for a particular morning but remembered that the two regression lines remain unchanged on its inclusion. Find this missing observation. [1] Mr. Lee needs to report to work by 8.00 am. He leaves home x minutes after 6.00 am and he arrives at his workplace z minutes before 8.00 am. (iii) Find z in terms of x. [1] (iv) Hence estimate, to the nearest minutes, the latest time Mr. Lee can leave home without arriving late at work. [2] 8 A website about electric motors gives information about the percentage efficiency of motors depending on their power, measured in horsepower. Xian has copied the following table for a particular type of electric motor,