1. The Type Fast secretarial training agency has a new computer software spreadsheet package. The agency investigates the number of hours it takes people of varying ages to reach a level of proficiency using this package. Fifteen individuals are tested and the results are summarised in the table below. Age (x) 32 40 21 45 24 19 17 21 27 54 33 37 23 45 18 Time (in hours) (y) 10 12 8 15 7 8 6 9 11 16 t 13 9 17 5 (a) (i) Given that Sy = 3.5 and Sxy = 36.7, calculate the product-moment correlation coefficient r for this data. (4) (ii) What does the value of the correlation coefficient suggest about the relationship between the two variables? (1) (b) Given that the mean time taken was 10.6 hours, write the equation of the regression line for y on x in the form y = ax + b. (3) (c) Use your equation for the regression line to predict (i) the time that it would take a 33 year old person to reach proficiency, giving your answer correct to the nearest hour; (2) (ii) the age of a person who would take 8 hours to reach proficiency, giving your answer correct to the nearest year. (2) (Total 12 marks) IB Questionbank Mathematical Studies 3rd edition 1 2. Ten students were given two tests, one on Mathematics and one on English. The table shows the results of the tests for each of the ten students. Student A Mathematics (x) 8.6 English (y) 33 (a) B C 13.4 12.8 51 30 D E F G H I J 9.3 1.3 9.4 13.1 4.9 13.5 9.6 48 12 23 46 18 36 50 Given sxy (the covariance) is 35.85, calculate, correct to two decimal places, the product moment correlation coefficient (r). (6) (b) Use your result from part (a) to comment on the statement: ‘Those who do well in Mathematics also do well in English.’ (2) (Total 8 marks) IB Questionbank Mathematical Studies 3rd edition 2 3. Ten students were asked for their average grade at the end of their last year of high school and their average grade at the end of their last year at university. The results were put into a table as follows: (a) Studen t High School grade, x University grade, y 1 2 3 4 5 6 7 8 9 10 90 75 80 70 95 85 90 70 95 85 3.2 2.6 3.0 1.6 3.8 3.1 3.8 2.8 3.0 3.5 Total 835 30.4 Given that sx = 8.96, sy = 0.610 and sxy = 4.16, find the correlation coefficient r, giving your answer to two decimal places. (2) (b) Describe the correlation between the high school grades and the university grades. (2) (c) Find the equation of the regression line for y on x in the form y = ax + b. (2) (Total 6 marks) IB Questionbank Mathematical Studies 3rd edition 3 4. Several candy bars were purchased and the following table shows the weight and the cost of each bar. Weight (g) Cost (Euros) (a) Yummy Chox Marz Twin Chunx Lite BigC Bite 60 85 80 65 95 50 100 45 1.10 1.50 1.40 1.20 1.80 1.00 1.70 0.90 Given that sx = 19.2, sy = 0.307 and sxy = 5.81, find the correlation coefficient, r, giving your answer correct to 3 decimal places. (2) (b) Describe the correlation between the weight of a candy bar and its cost. (1) (c) Calculate the equation of the regression line for y on x. (3) (d) Use your equation to estimate the cost of a candy bar weighing 109 g. (2) (Total 8 marks) IB Questionbank Mathematical Studies 3rd edition 4 5. A number of employees at a factory were given x additional training sessions each. They were then timed on how long (y seconds) it took them to complete a task. The results are shown in the scatter diagram below. A list of descriptive statistics is also given. 14 time taken (seconds) 12 10 8 6 4 2 0 2 4 6 8 10 number of additional training sessions n = 9, sum of x values: x = 54, sum of y values: y = 81, mean of x values: x = 6, mean of y values: y = 9, standard deviation of x: sx = 1.94, standard deviation of y: sy = 2.35, covariance: sxy = –3.77. (a) Determine the product-moment correlation coefficient (r) for this data. (2) (b) What is the nature of the relationship between the amount of additional training and the time taken to complete the task? (2) n (c) Calculate (x i 1 i – x )( yi – y ) given that the covariance sxy = –3.77. (1) (d) (i) (ii) Determine the equation of the linear regression line for y on x. Find the expected time to complete the task for an employee who only attended three additional training sessions. (4) (Total 9 marks) IB Questionbank Mathematical Studies 3rd edition 5 Solutions 1. (a) (i) Sx = 11.2 36.7 r= (M2) 11.2 3.5 = 0.936 (3 s.f.) (A1) (A1) OR Sx = 11.6 36.7 r= 11.6 3.5 = 0.904 (3 s.f.) (A1) (M2) (A1) (ii) The correlation coefficient suggests a strong positive correlation between the two variables. (R1) _ (b) (c) _ Sxy ( x x ) ( Sx) 2 36.7 y – 10.6 = (x – 30.4) 11.2 2 y = 0.293x + 1.69 (or y = 0.293x + 1.71) Allow ft from (a) (i)) 5 y– y (i) (ii) (M1) (A2) y = 0.293 × 33 + 1.69 = 11.359 = 11 hours (M1) 8 = 0.293x + 1.69 x = 21.54 = 22 years (M1) 3 (A1) (A1) 4 [12] 2. (a) For applying r = S xy S x S y or any correct formulae. For: Sx = 4.0034568… 4 (M2) Sy = 13.992456… 14 (M2) r = 0.6399706… 0.64 (2 d.p.) (A1) (M1) 6 Note: Follow through with candidate’s Sx + Sy. Accept solutions that use the unbiased estimates for the population standard deviations. (b) This indicates that there is a degree of positive correlation between scores in Mathematics and scores in English. (R1) Note: Follow through using candidate’s v,2 and 52% (v) from table. Therefore those who do well in Mathematics are likely to do well in English also. (Or equivalent statements.) (R1) 2 [8] IB Questionbank Mathematical Studies 3rd edition 6 3. (a) r= S xy SxSy 4.16 (8.96)(0.610) = 0.76 = (M1) (A1) 2 (b) There is a fairly strong positive correlation between high school grades and university grades. (A1) (A1) 2 Note: Award (A1) for strong (or fairly strong) or high, (A1) for positive. (c) y– y _ S xy S x2 _ ( x x) 4.16 (x – 83.5) 8.96 2 y = 0.052x – 1.29 (3 s.f.) Note: Award (C2) for correct answer (from calculator). y – 3.04 = (M1) (A1) 2 [6] 4. (a) r= S xy (S x S y ) 5.81 (19.2 0.307) = 0.986 (M1) (A1) 2 1 Note: Award (G2) for 0.985 from GDC. (b) Strong, positive correlation (A1) (c) y = 0.182 + 0.0158x (G3) OR 5.81 (x – 72.5) 19.2 2 y = 0.0158x + 0.182 y – 1.325 = (d) y = 0.0158 × 109 + 0.182 = 1.90 euros. (M1)(A1) (A1) 3 (M1) (A1) 2 [8] 5. (a) r= 3.77 = –0.827 1.94 2.35 or (G2) (b) (M1)(A1) 2 moderate/strong (allow approximately linear) (A1) negative (A1) 2 Note: Comments such as: number of sessions increases as time decreases can be awarded (A1)(A0) but “inversely proportional” receives no marks. IB Questionbank Mathematical Studies 3rd edition 7 n (c) (x i x)( y i y ) = –3.77 × 9 = –33.9 (A1) 1 i 1 (d) (i) (y – 9) = 3.77 (x – 6) 1.94 2 y = –x + 15 (ii) –3 + 15 = 12 seconds (M1) (A1) or (G2) (M1)(A1) or (G2) 4 [9] IB Questionbank Mathematical Studies 3rd edition 8

© Copyright 2020