EPMATH235 Statistics Extra Exercises Worksheet PDF

EPMATH235 STATISTICS – Extra exercises WORKSHEET Module 8/Lecture 8 –Numerical bivariate data 1. For each pair of variables, state whether they have a positive, negative or no correlation. (a) height and shoe size of an individual (b) star sign and happiness (...

EPMATH235 STATISTICS – Extra exercises WORKSHEET Module 8/Lecture 8 –Numerical bivariate data 1. For each pair of variables, state whether they have a positive, negative or no correlation. (a) height and shoe size of an individual (b) star sign and happiness (c) age and hearing ability (d) exam scores and time spent on homework. 2. The scores awarded by two judges to a sample of competitors at a diving competition are shown in the table. Competitor 1 2 3 4 5 6 7 8 9 10 Judge A 5 6.5 8 9 4 2.5 7 5 6 3 Judge B 6 7 8.5 9 5 4 7.5 5 7 4.5 (a) Calculate the correlation coefficient r (b) Comment on the consistency or otherwise of the scores awarded by the two judges. Solving for x and y: Module8/Lecture 8 In this module, we found the equation of the Least-Squares Regression line. From this equation, you are then asked to solve for x and solve for y. Say that the equation of the regression line is in the general form : 𝑦𝑦 = 𝑏𝑏 𝑥𝑥 + 𝑎𝑎 We found (using the STAT mode) that the value of 𝑏𝑏 = −2.5 and 𝑎𝑎 = 5. 10 Given these values of 𝑏𝑏 and 𝑎𝑎 the equation of the regression line is : 𝑦𝑦 = −2.5𝑥𝑥 + 10 Using this equation, we are then asked to make predictions. Let us start with x = 10, what will 𝑦𝑦 be? First, substitute x = 10 into the equation. 𝑦𝑦 = (−2.5 × 10) + 10 = −25 + 510 = −2015 Now we have predicted that when x = 10, 𝑦𝑦 = −20 15 Now, let us have an example of solving for x when 𝑦𝑦 = 100. Here we will need to have a basic knowledge of algebra to solve for x. First, substitute 𝑦𝑦 = 100 into the equation. 100 = −2.5𝑥𝑥 + 10 1 Now we to solve for x we have to have the unknown x on one side of the equal sign and all the known values (numbers) on the other side of the equal sign. To do this, let us move the +10 to the other side of the equal sign. To do this I have to subtract 10 from both sides of the equal sign. In doing this operation, we remove the +10 from the right-hand side, leaving us with -2.5x. 100 − 10 = −2.5𝑥𝑥 + 10 − 10 90 = −2.5𝑥𝑥 Now I need to divide both sides by -2.5. This operation will now leave us with x on the right- hand side. 90 −2.5𝑥𝑥 = −2.5 −2.5 −38 - 36 = 𝑥𝑥 Now we estimate that when y = 100, x = -38. - 36 Alternatively, if you rearrange the general form of the regression line to this: 𝑦𝑦 = 𝑏𝑏 𝑥𝑥 + 𝑎𝑎 − 𝒂𝒂 𝒚𝒚 𝒙𝒙 = 𝒃𝒃 You can put the values of 𝑏𝑏 and 𝑎𝑎 into this form and solve for x. Here are a few exercises you can practice with (answers at the end). For the following equations, solve for y and x. 1. 𝑦𝑦 = 12𝑥𝑥 − 10 : a. solve for y when x = 7, b. solve for x when y = 50 2. 𝑦𝑦 = −4𝑥𝑥 − 15 : a. solve for y when x = -10, b. solve for x when y = 90 3. 𝑦𝑦 = −0.09𝑥𝑥 + 58 : a. solve for y when x = 300, b. solve for x when y = 50 4. 𝑦𝑦 = 1.73x+ 125 a. solve for y when x = 29, b. solve for x when y = 180 Answers: 1. a. y = 74 b. x = 5 2. a. y = 25 b. x = -26.25 2 3. a. y = 31 b. x = 88.9 4. a. y = 175.17 b. x = 31.79 3. During a medical check, the systolic and diastolic blood pressure (in millimetres of mercury) were recorded for a group of patients in one ward of a hospital. The following results were obtained: Systolic 100 123 172 150 110 120 125 195 112 105 135 130 Diastolic 65 80 95 90 75 78 75 100 70 65 80 82 (a) Determine the equation of the least squares’ regression line. (b) Predict the diastolic blood pressure for a person with systolic blood pressure of 192 millimetres of mercury, using the equation in part (a). 4. Jerry has set up a small business selling his home-made Sushi to the college canteen. He analysed the relationship between price and demand per week, for several weeks. The information is given below. Price P ($) 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 Demand D 40 40 35 33 29 28 24 23 (a) Calculate the correlation co-efficient r (correct to 2 d.p.) (b) Determine the equation of the least squares regression line. (c) Use the equation to determine the demand (nearest whole number) for Sushi if the price is (i) $1.25 (ii) $2. (d) Jerry decides to limit the demand to 34 units per week. Use the equation to determine the price he should sell the Sushi. (nearest 5 cent) 5. Data was collected on the heights and weights of 12 students to see if there was a relationship. Height(cm) 170 158 150 164 177 186 173 190 156 180 160 187 Weight(kg) 68 55 53 58 67 81 71 78 59 80 67 73 (a) Find the correlation coefficient (to 3 d.p.) (b) Describe the relationship between the heights and weights of the students. (c) Determine the equation of the least squares regression line. (d) Use the equation in (c) to estimate the weight of a student whose height was 163cm. 6. The amounts of energy, E(kJ), used per day by 8 men of various weights, W(kg),are shown in the table. Weight, W(kg) 78 89 92 95 101 115 118 125 Energy, E(kJ) 1610 1820 1650 2030 1860 1970 2100 2340 a) Find the correlation coefficient (to 3 d.p) b) Determine the equation of the least squares regression line. c) Predict the amount of energy used by a man weighing 99kg. d) Use the equation to estimate the weight of a man that uses 2200kJ per day. 3 EPMATH235 STATISTICS – Extra exercises WORKSHEET - Solutions Module 8/Lecture 8 –Numerical bivariate data Answers: 1. (a) positive correlation (as height increases, foot size usually increases) (b) no correlation (c) negative correlation (as age increases, hearing ability usually decreases) (d) positive correlation (as time spent on homework increases, exam scores usually increase) 2. a) r = 0.98 (2d.p.) b) Consistency between judges in awarding scores is high. 3. (a) 𝑦𝑦 = 0.37𝑥𝑥 + 30.57 (b) 102.0 millimetres of mercury (1 d.p.) 4. (a) r = −0.99 (b) 𝐷𝐷 = −26.67𝑃𝑃 + 70.17 (c) (i) 37 (ii) 17 (d) $1.35 5. (a) r= 0.897 (b) strong positive relationship (as heights increase, so do weights) (c) y= 0.65x – 42.96 (d) 62.99kg 6. (a) r= 0.874 (b) y=12.95x + 606.59 (c) 1889kJ (to nearest whole) (d) 123kg (to nearest whole) 4