Arab Evangelical Episcopal Church Council 12AASL (ASG & BSA) Name Statistics Feb 2022 1a. [2 marks]A large company surveyed 160 of its employees to find out how much time they spend traveling to work on a given day. The results of the survey are shown in the following cumulative frequency diagram. Find the median number of minutes spent traveling to work. 1b. [3 marks]Find the number of employees whose travelling time is within 15 minutes of the median. 1c. [3 marks]Only 10% of the employees spent more than π minutes traveling to work. Find the value of π. 1d. [1 mark]The results of the survey can also be displayed on the following box-andwhisker diagram. Write down the value of π. 1e. [2 marks]Find the value of π. 1f. [2 marks]Hence, find the interquartile range. 1g. [2 marks]Travelling times of less than π minutes are considered outliers.Find the value of π. 2a. [2 marks]The following table below shows the marks scored by seven students on two different mathematics tests. Let L1 be the regression line of x on y. The equation of the line L1 can be written in the form x = ay + b. Find the value of a and the value of b. 2b. [3 marks]Let L2 be the regression line of y on x. The lines L1 and L2 pass through the same point with coordinates (p , q). Find the value of p and the value of q. 3. [4 marks]A data set consisting of 16 test scores has mean 14.5 . One test score of 9 requires a second marking and is removed from the data set. Find the mean of the remaining 15 test scores. 4a. [2 marks]The following table shows the systolic blood pressures, π mmHg, and the ages, π‘ years, of 6 male patients at a medical clinic. Determine the value of Pearson’s productβmoment correlation coefficient, π, for these data. 4b. [1 mark ] Interpret, in context, the value of π found in part (a) (i). 4c. [2 marks] The relationship between π‘ and π can be modelled by the regression line of π on π‘ with equation π = ππ‘ + π . Find the equation of the regression line of π on π‘. 4d. [2 marks] A 50βyearβold male patient enters the medical clinic for his appointment. Use the regression equation from part (b) to predict this patient’s systolic blood pressure. 4e. [1 mark ]A 16βyearβold male patient enters the medical clinic for his appointment. Explain why the regression equation from part (b) should not be used to predict this patient’s systolic blood pressure. 5a. [1 mark]The principal of a high school is concerned about the effect social media use might be having on the self-esteem of her students. She decides to survey a random sample of 9 students to gather some data. She wants the number of students in each grade in the sample to be, as far as possible, in the same proportion as the number of students in each grade in the school. State the name for this type of sampling technique. 5b. [3 marks]The number of students in each grade in the school is shown in table. Show that 3 students will be selected from grade 12. 5c. [2 marks]Calculate the number of students in each grade in the sample. 5d. [1 mark]In order to select the 3 students from grade 12, the principal lists their names in alphabetical order and selects the 28th, 56th and 84th student on the list. State the name for this type of sampling technique. 5e. [2 marks]Once the principal has obtained the names of the 9 students in the random sample, she surveys each student to find out how long they used social media the previous day and measures their self-esteem using the Rosenberg scale. The Rosenberg scale is a number between 10 and 40, where a high number represents high self-esteem. Calculate Pearson’s product moment correlation coefficient, π. 5f. [1 mark]Interpret the meaning of the value of π in the context of the principal’s concerns. 5g. [1 mark]Explain why the value of π makes it appropriate to find the equation of a regression line. 5h. [4 marks]Another student at the school, Jasmine, has a self-esteem value of 29. By finding the equation of an appropriate regression line, estimate the time Jasmine spent on social media the previous day. 6a. [3 marks]A research student weighed lizard eggs in grams and recorded the results. The following box and whisker diagram shows a summary of the results where πΏ and π are the lower and upper quartiles respectively. The interquartile range is 20 grams and there are no outliers in the results. Find the minimum possible value of π. 6b. [2 marks]Hence, find the minimum possible value of πΏ. 7a. [1 mark] The fastest recorded speeds of eight animals are shown in the following table. State whether speed is a continuous or discrete variable. 7b. [1 mark]Write down the median speed for these animals. 7c. [1 mark]Write down the range of the animal speeds. 7d. [2 marks]For these eight animals find the mean speed. 7e. [1 mark]For these eight animals write down the standard deviation. 8a. [2 marks]A group of 10 girls recorded the number of hours they spent watching television during a particular week. Their results are summarized in the box-and-whisker plot below. The range of the data is 16. Find the value of π. 8b. [2 marks]Find the value of the interquartile range. 8c. [2 marks]The group of girls watched a total of 180 hours of television. Find the mean number of hours that the girls in this group spent watching television that week. 8d. [2 marks]A group of 20 boys also recorded the number of hours they spent watching television that same week. Their results are summarized in the table below. Find the total number of hours the group of boys spent watching television that week. 8e. [3 marks]Find the mean number of hours that all 30 girls and boys spent watching television that week. 8f. [2 marks]The following week, the group of boys had exams. During this exam week, the boys spent half as much time watching television compared to the previous week. For this exam week, find the mean number of hours that the group of boys spent watching television. 9a. [1 mark]University students were surveyed and asked how many hours, β , they worked each month. The results are shown in the following table. Use the table to find the following values.π. 9b. [1 mark]π. 9c. [2 marks]The first five class intervals, indicated in the table, have been used to draw part of a cumulative frequency curve as shown. On the same grid, complete the cumulative frequency curve for these data. 9d. [2 marks]Use the cumulative frequency curve to find an estimate for the number of students who worked at most 35 hours per month. Markscheme 1a. [2 marks]evidence of median position (M1) 80th employee 40 minutes A1 [2 marks] 1b. [3 marks]valid attempt to find interval (25–55) 18 (employees), 142 (employees) 124 (M1) A1 A1 [3 marks] 1c. [3 marks]recognising that there are 16 employees in the top 10% 144 employees travelled more than π minutes π = 56 A1 [3 marks] 1d. [1 mark] π = 70 A1 [1 mark] 1e. [2 marks] recognizing π is first quartile value 40 employees π = 33 A1 [2 marks] 1f. [2 marks] 47 − 33 (M1) (M1) (A1) (M1) IQR = 14 A1 1g. [2 marks]attempt to find 1.5 × their IQR (M1) 33 − 21 12 (A1) 2a. [2 marks] a = 1.29 and b = −10.4 A1A1 2b. recognising both lines pass through the mean point p = 28.7, q = 30.3 (M1) A2 3. * This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers. 16 π΄ π₯π π=1 16 (M1) = 14.5 π Note: Award M1 for use of π₯ = 16 ⇒ π΄ π₯π = 232 (A1) 232−9 (A1) π=1 new π₯ = 15 = 14.9 (= 14.86, = 223 15 ) π΄ π₯π π=1 π . A1 Note: Do not accept 15. 4a.* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers. π = 0.946 A2 4b. the value of π shows a (very) strong positive correlation between age and (systolic) blood pressure A1 4c. π = 1.05π‘ + 69.3 A1A1 Note: Only award marks for an equation. Award A1 for π = 1.05 and A1 for π = 69.3. Award A1A0 for π¦ = 1.05π₯ + 69.3. 4d. 122 (mmHg) (M1)A1 4e. the regression equation should not be used because it involves extrapolation A1 5aStratified sampling A1 [1 mark] 5b. There are 260 students in total 84 260 A1 M1A1 × 9 = 2.91 So 3 students will be selected. 60 AG 83 33 5c. grade 9 = 260 × 9 ≈ 2, grade 10 = 260 × 9 ≈ 3, grade 11 = 260 × 9 ≈ 1 5d. Systematic sampling 5e. π = −0.901 A2 A1 A2 5f. [1 mark] The negative value of π indicates that more time spent on social media leads to lower selfesteem, supporting the principal’s concerns. R1 [1 mark] 5g. π being close to –1 indicates there is strong correlation, so a regression line is appropriate. R1 [1 mark] 5h. [4 marks] Find the regression line of π‘ on π . π‘ = −0.281π + 9.74 M1 A1 π‘ = (−0.2807 … )(29) + 9.739 … = 1.60 hours M1A1 [4 marks] 6a. attempt to use definition of outlier 1.5 × 20 + π3 (M1) 1.5 × 20 + π ≥ 75 (⇒ π ≥ 45, accept π > 45) OR 1.5 × 20 + π3 = 75 minimum value of π = 45 A1 A1 6b. [attempt to use interquartile range (M1) π − πΏ = 20 (may be seen in part (a)) OR πΏ ≥ 25 (accept πΏ > 25) minimum value of πΏ = 25 A1 7a. * This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure. continuous (A1) (C1) 7b. 75.5 (km h−1) (A1) (C1) Note: Answer must be exact. 7c. 294 (km h−1) (A1) (C1) [1 mark] 7d. [2 marks] 300+97+80+80+71+64+21+6 8 OR 719 8 (M1) Note: Award (M1) for correct sum divided by 8. 89.9 (89.875)(km h−1) (A1) (C2) [2 marks] 7e. [1 mark] 84.6 (84.5597…)(km h−1) (A1) (C1) Note: If the response to part (d)(i) is awarded zero marks, a correct response to part (d)(ii) is awarded (C2). [1 mark] 8a. [2 marks] * This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure. valid approach (M1) eg 16 + 8, π − 8 24 (hours) A1 N2 [2 marks] 8b. [2 marks] valid approach (M1) eg 20 − 15, π3 − π1, 15 − 20 IQR = 5 A1 N2 [2 marks] 8c. [2 marks] correct working eg (A1) 180 180 ∑π₯ 10 , π , 10 mean = 18 (hours) A1 N2 [2 marks] 8d. [2 marks] attempt to find total hours for group B (M1) eg π₯ × π group B total hours = 420 (seen anywhere) [2 marks] A1 N2 8e. [3 marks] attempt to find sum for combined group (may be seen in working) eg 180 + 420, 600 correct working eg (A1) 180+420 600 , 30 30 mean = 20 (hours) A1 N2 [3 marks] 8f. [2 marks] valid approach to find the new mean eg 1 2 π, mean = 1 (M1) × 21 2 21 2 (= 10.5) hours A1 N2 [2 marks] 9a. [1 mark] π = 10 (A1) (C1) Note: Award (A1) for each correct value. [1 mark] 9b. [1 mark] Markscheme π = 56 (A1) (C1) Note: Award (A1) for each correct value. [1 mark] 9c. [2 marks] (M1) Markscheme (A1)(A1) (C2) Note: Award (A1)(ft) for their 3 correctly plotted points; award (A1)(ft) for completing diagram with a smooth curve through their points. The second (A1)(ft) can follow through from incorrect points, provided the gradient of the curve is never negative. Award (C2) for a completely correct smooth curve that goes through the correct points. [2 marks] 9d. [2 marks a straight vertical line drawn at 35 (accept 35 ± 1) (M1) 26 (students) (A1) (C2) Note: Accept values between 25 and 27 inclusive. [2 marks]