S1 Revision Pack 1 Averages, Spread & Diagrams Doublestruck & CIE - Licensed to Brillantmont International School 1
1. A group of 10 married couples and single men found that the mean age of the 10 women was 1. years and the standard deviation of the women s ages was 1.1 years. x w For the 1 men, the mean age was. years and the standard deviation was 1.7 years. x m (i) (ii) Find the mean age of the whole group of people. The individual women s ages are denoted by x w and the individual men s ages by x m. By w m first finding x and x, find the standard deviation for the whole group. [] [] Doublestruck & CIE - Licensed to Brillantmont International School
. Rachel measured the lengths in millimetres of some of the leaves on a tree. Her results are recorded below. 7 8 9 Find the mean and standard deviation of the lengths of these leaves. [] Doublestruck & CIE - Licensed to Brillantmont International School
. A summary of observations of x gave the following information: The mean of these values of x is 8.9. Σ(x a) = 7. and Σ(x a) = 11. (i) Find the value of the constant a. (ii) Find the standard deviation of these values of x. [] [] Doublestruck & CIE - Licensed to Brillantmont International School
. The length of time, t minutes, taken to do the crossword in a certain newspaper was observed on 1 occasions. The results are summarised below. Σ(t ) = 1 Σ(t ) = 8. Calculate the mean and standard deviation of these times taken to do the crossword. [] Doublestruck & CIE - Licensed to Brillantmont International School
. The following table shows the results of a survey to find the average daily time, in minutes, that a group of schoolchildren spent in internet chat rooms. Time per day (t minutes) Frequency 0 t < 10 10 t < 0 f 0 t < 0 11 0 t < 80 The mean time was calculated to be 7. minutes. (i) (ii) Form an equation involving f and hence show that the total number of children in the survey was. Find the standard deviation of these times. [] [] Doublestruck & CIE - Licensed to Brillantmont International School
. As part of a data collection exercise, members of a certain school year group were asked how long they spent on their Mathematics homework during one particular week. The times are given to the nearest 0.1 hour. The results are displayed in the following table. Time spent (t hours) 0.1 t 0. 0. t 1.0 1.1 t.0.1 t.0.1 t. Frequency 11 1 18 0 1 (i) (ii) Draw, on graph paper, a histogram to illustrate this information. Calculate an estimate of the mean time spent on their Mathematics homework by members of this year group. [] [] Doublestruck & CIE - Licensed to Brillantmont International School 7
7. The lengths of time in minutes to swim a certain distance by the members of a class of twelve 9-year-olds and by the members of a class of eight 1-year-olds are shown below. 9-year-olds: 1.0 1.1 1.0 1. 1.9 1.1 1. 1.7 1.7 1. 1.0 1. 1-year-olds: 1.8 1.0 11. 11.7 1. 1.7 1.8 1.9 (i) (ii) Draw a back-to-back stem-and-leaf diagram to represent the information above. A new pupil joined the 1-year-old class and swam the distance. The mean time for the class of nine pupils was now 1. minutes. Find the new pupil s time to swim the distance. [] [] Doublestruck & CIE - Licensed to Brillantmont International School 8
8. Each father in a random sample of fathers was asked how old he was when his first child was born. The following histogram represents the information. Frequency density (fathers per year) 1 0 0 10 0 0 0 0 0 70 Age (years) (i) (ii) (iii) (iv) What is the modal age group? How many fathers were between and 0 years old when their first child was born? How many fathers were in the sample? Find the probability that a father, chosen at random from the group, was between and 0 years old when his first child was born, given that he was older than years. [1] [] [] [] Doublestruck & CIE - Licensed to Brillantmont International School 9
9. The weights of 0 children in a class, to the nearest kilogram, were as follows. 0 1 7 9 1 0 7 7 9 1 8 0 1 9 8 Construct a grouped frequency table for these data such that there are five equal class intervals with the first class having a lower boundary of 1. kg and the fifth class having an upper boundary of 1. kg. [] Doublestruck & CIE - Licensed to Brillantmont International School 10
10. A study of the ages of car drivers in a certain country produced the results shown in the table. Percentage of drivers in each age group Young Middle-aged Elderly Males 0 Females 0 70 10 Illustrate these results diagrammatically. [] Doublestruck & CIE - Licensed to Brillantmont International School 11
11. The pulse rates, in beats per minute, of a random sample of 1 small animals are shown in the following table. 11 10 18 1 1 10 1 10 1 10 1 117 109 1 1 (i) (ii) (iii) Draw a stem-and-leaf diagram to represent the data. Find the median and the quartiles. On graph paper, using a scale of cm to represent 10 beats per minute, draw a box-and-whisker plot of the data. [] [] [] Doublestruck & CIE - Licensed to Brillantmont International School 1
1. The following back-to-back stem-and-leaf diagram shows the cholesterol count for a group of people who exercise daily and for another group of who do not exercise. The figures in brackets show the number of people corresponding to each set of leaves. (9) (1) (9) (7) () () (1) (0) People who exercise 9 8 9 8 8 8 7 8 7 7 7 7 9 8 1 1 1 7 8 9 10 People who do not exercise 1 1 1 1 1 7 7 8 8 7 7 9 8 9 7 8 8 9 7 7 8 9 9 8 9 () () (1) (1) (9) (9) () () (i) (ii) Key: 8 1 represents a cholesterol count of 8. in the group who exercise and 8.1 in the group who do not exercise Give one useful feature of a stem-and-leaf diagram. Find the median and the quartiles of the cholesterol count for the group who do not exercise. [1] [] You are given that the lower quartile, median and upper quartile of the cholesterol count for the group who exercise are.,. and. respectively. (iii) On a single diagram on graph paper, draw two box-and-whisker plots to illustrate the data. [] Doublestruck & CIE - Licensed to Brillantmont International School 1
1. During January the numbers of people entering a store during the first hour after opening were as follows. Time after opening, x minutes Frequency Cumulative frequency 0 < x 10 10 10 10 < x 0 1 0 < x 0 78 0 < x 0 7 a 0 < x 0 b 0 (i) Find the values of a and b. [] (ii) (iii) (iv) Draw a cumulative frequency graph to represent this information. Take a scale of cm for 10 minutes on the horizontal axis and cm for 0 people on the vertical axis. Use your graph to estimate the median time after opening that people entered the store. Calculate estimates of the mean, m minutes, and standard deviation, s minutes, of the time after opening that people entered the store. [] [] [] (v) Use your graph to estimate the number of people entering the store between 1 and m s minutes after opening. 1 m s [] Doublestruck & CIE - Licensed to Brillantmont International School 1
1. The arrival times of 0 trains were noted and the number of minutes, t, that each train was late was recorded. The results are summarised in the table. Number of minutes late (t) t < 0 0 t < t < t < t < 10 Number of trains 1 9 19 (i) (ii) Explain what t < 0 means about the arrival times of trains. Draw a cumulative frequency graph, and from it estimate the median and the interquartile range of the number of minutes late of these trains. [1] [7] Doublestruck & CIE - Licensed to Brillantmont International School 1
1. In a survey, people were asked how long they took to travel to and from work, on average. The median time was hours minutes, the upper quartile was hours minutes and the interquartile range was hours 8 minutes. The longest time taken was hours 1 minutes and the shortest time was 0 minutes. (i) (ii) Find the lower quartile. Represent the information by a box-and-whisker plot, using a scale of cm to represent 0 minutes. [] [] Doublestruck & CIE - Licensed to Brillantmont International School 1
1. The stem-and-leaf diagram below represents data collected for the number of hits on an internet site on each day in March 007. There is one missing value, denoted by x. 0 1 0 1 1 1 1 1 7 x 9 8 8 9 8 9 () () (9) (7) () Key: 1 represents 1 hits (i) Find the median and lower quartile for the number of hits each day [] (ii) The interquartile range is 19. Find the value of x. [] Doublestruck & CIE - Licensed to Brillantmont International School 17
17. The salaries, in thousands of dollars, of 11 people, chosen at random in a certain office, were found to be: 0,,, 1,, 0, 0, 8, 1, 9, 7. Choose and calculate an appropriate measure of central tendency (mean, mode or median) to summarise these salaries. Explain briefly why the other measures are not suitable. [] Doublestruck & CIE - Licensed to Brillantmont International School 18
Solutions Doublestruck & CIE - Licensed to Brillantmont International School 19
1. (i) (1. 10 +. 1) / For multiplying by 10 and 1 respctively and dividing by =.1 A1 For correct answer (ii) 1.1 xw = 1. 10 For correct substitution from recognisable formula with or without sq rt x w = 19. For correct x w m (can be rounded) 1.7 x =. A1 1 x m = 99.7 A1 For correct x m (can be rounded) Total Σ = 919. For using and their answer to (i) in correct formula 919. Sd =.1 = 1.0 A1 For correct answer [7]. mean = 8. mm Correct answer Correct method if shown (can be implied) must see a sign sd =.7 mm c.a.o Correct answer. (i) 7./ (=.0) Accept ( 7. + anything)/ OR a = 8.9 +.0 = 1 Correct answer 8.9 (= 1.8) Σx Σa = 7. For 8.9 seen Σa = 88 a = 1 A1 A1 A1 [] Doublestruck & CIE - Licensed to Brillantmont International School 0
Correct answer obtained using Σx and Σa 11 (ii) standard deviation = (.0) 11 For ( ± their coded mean) OR = 8.88 A1 Correct answer 81. sd = 8.9 their For x 8.9 where Σx is obtained from expanding Σ(x a) with aσx seen = 8.88 A1 Correct answer []. mean = 1/1 For 1/1 seen =.7 (.8) minutes A1 Correct answer sd = 8. /1 ( 1 /1) 8./1 ( ± their coded mean) =. minutes A1 Correct answer. (i) + 1f + 0 11 + 0 = 7.(17 + f) For attempt at LHS, accept end points or cl width [] For attempt at RHS, must have 17+ f f = 9 For correct f total = AG For correct answer given, ft if previous answer rounds to 9 A1 A1 (ii) σ = 1.1 For method including sq rt and mean squared (can be implied if using calculator, Doublestruck & CIE - Licensed to Brillantmont International School 1
must be x f on mid-points) or f ( x x) For correct answer A1 []. (i) Fd:, 0, 18, 0, 1 Attempt at freq density or scaling fd 0 0 10 0 1 time Bar lines correctly located at 0., 1.0,.0,.0, no gaps correct widths of bars both axes uniform from at least 0 to 1 or 0, and 0.0 to. and labelled, (fd, or freq per half hour, time, hours, t) (ii) mid-points 0., 0.8, 1.,.,.8 an attempt at mid-points (not class widths) = 199. / 9 using (Σ their fx) / their 9 7. (i) mean =.1 hours correct answer from 199. in num 1 yr olds 9 year olds 7, 11 9, 8, 1 7, 0 1 0,, 7, 8 1,, A1 [8] Doublestruck & CIE - Licensed to Brillantmont International School
1 0, 1, 9, 1 0, 1,, 7, columns including an integer stem in the middle, single digits in leaves. Can go downwards One leaf column correct, ordering not necessary Other leaf column correct (ordering not nec) and both leaves labelled correctly (could be in key) Key 7 1 means 1.7 minutes and 1. minutes Key correct both ways or two keys one each way, must have minutes (ii) (8 pupils) = 10.8 10.8 seen or implied for 1. 9 (9 pupils) = 1. 9 (= 1.) New pupil s time = 1. min Ft on 1. their 8 8. (i) 0- years 1 ft [7] (ii).8 Multiplying by = A1 Correct answer (iii) + 18 + + 8 + +10 Summing their attempts at frequencies = 110 A1 Correct answer (iv) /88 Dividing their (ii) by their attempt at > group = 0.7 A1 ft Correct answer, ft on above [7] Doublestruck & CIE - Licensed to Brillantmont International School
Doublestruck & CIE - Licensed to Brillantmont International School
9. Weight freq 1.-..-9. 7 9.-. 10.-7. 7.-1. Five groups Correct boundaries, accept -, -9 etc Attempt to calculate frequencies Σ 9, 0 or 1. frequencies correct A1 A1 [] 10. two pie charts or bars (m and f) lots of or lots of, bars, lines or sectors 11. (i) different age categories in each group one category touching, not superimposed, one category not touching, bars equal width correct height or angle accept pie chart visually correct labels m and f, percentage, drivers, y,m elderly A1 [] 10 11 1 1 1 1 1 9 7 0 8 0 8 Correct stem Correct leaves, must be sorted and in columns and give correct overall shape Doublestruck & CIE - Licensed to Brillantmont International School
key represents 10 10 Key, must have vertical line in both (ii) median = 1 Any correct values seen LQ = 11 UQ = 1 third correct value (iii) 100 110 10 10 10 10 10 170 pulse rate correct uniform scale from at least 110 to 10 with room for end points, and label or title correct median and quartiles on diagram ft their values (must be box ends) correct whiskers, no line through box, touching box in the middle not the top or bottom 1. (i) shows all the data 1 Or other suitable advantage e.g. can see the shape, mode etc. ft [8] Doublestruck & CIE - Licensed to Brillantmont International School
(ii) Not exercise LQ =. Median =. ft UQ = 8. ft ft on first answer missing the decimal point (iii) not ex ex 7 8 9 10 For one linear numbered scale from to 9., or two identically positioned scales For not exercise all correct on linear scale For exercise correct on linear scale For two labels and cholesterol and scale labelled ft SR non linear scale max B0 B0 B0 SR no graph paper lose one mark [8] 1. (i) a = 9 b = (ii) Doublestruck & CIE - Licensed to Brillantmont International School 7
Correct linear scale minimum 0 to 0 and 0 to 0 Labels (cf or people or number of people) and (time, or minutes) and attempt at cf or cf step polygon Attempt to plot points at (10, 10), (0, ), (0, ), (0, 9) Correct graph through (0, 0) and (0, 0) A1 (iii) median is Attempt to read from graph at line y = 70 or 70. 1. to 1. min Correct answer A1 (iv) ( 10 + 1 1 + 78 + 7 + 0 ) / 0 = 980 / 0 Using mid points and frequencies = 18. min A1 Correct mean ( 10 + 1 1 +...) 18. Attempt at Σx f / Σf their mean numerically, could use cfs, ucb, but not class widths sd = 1. min Correct answer A1 (v) 18. ± 7.1 = 11.1,. 90 Attempt to read their mean ± 1 sd from cf graph = 1 to 170 people A1 Correct answer 1. (i) some trains were up to minutes early 1 Or sensible equivalent, must use the idea early not needed [1] (ii) cf table NB All M marks are independent Min Late, Less than 0 10 C freq 9 1 18 0 Doublestruck & CIE - Licensed to Brillantmont International School 8
cf 00 10 100 or curve 0 0 8 10 mins Attempt at C F table with upper limits no halves Uniform linear scales from at least 0 to 10 and 0 to 0 and at least one axis labelled, CF or mins or t Attempt at graph their points. (, 0) not nec (could be midpoints or lower bounds not f d) Attempt at median along 10 or 10. line Attempt at LQ along 1/ line and UQ along 1/1 line from graph Median = rounding to.1 to. min Correct median IQ range = rounding to. to. min Correct IQ range allow from midpoints etc 1. (i) LQ = hr min hr 8 min Subtracting IQR from UQ A17 [8] = min (0.9 hours) A1 Correct answer (ii) Doublestruck & CIE - Licensed to Brillantmont International School 9
0 1 time Correct whiskers (accept hour decimals or minutes) Correct median line, can be broken or extended Correct UQ and LQ ft on their (i), box ends correct uniform scale label hours or minutes, could be heading or key 1. (i) median = 1 th along = LQ = 1 not 1. ft [] (ii) UQ = LQ + 19 = For adding 19 to their LQ in whatever form x = Must be not. c.w.o. A1 [] 17. median $7000 Must have 7000 data have an outlier, are skew etc Accept any equivalent reason [] Doublestruck & CIE - Licensed to Brillantmont International School 0