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mathematics
cambridge international as & a level mathematics probability & statistics

Questions and Answers of Cambridge International AS & A Level Mathematics Probability & Statistics

a. The mean of 15, 31, 47, 83, 97, 119 and p2 is 63. Find the possible values of p.b. The mean of 6, 29, 3, 14, q, (q + 8), q2 and (10 – q) is 20. Find the possible values of q.
Twenty-five values of z are such that Σ(z − 7) = 275. Find z̅.
The total mass of gold extracted from a mine is denoted by Σx, which is measured in grams. Find an expression in x for the total mass in:a. Carats, given that 1 carat is equivalent to 200
a. Find the median for the values of t given in the following table.b. What feature of the data suggests that t is less than the median? Confirm whether or not this is the case. 7 9. 10 11 12 14 16 41
The mean mass of 13 textbooks is 875 grams, and n novels have a total mass of 13 706 grams. Find the mean mass of a novel, given that the textbooks and novels together have a mean mass of 716.6 grams.
Identify the mode of x and of y in the following tables. 5 6 -3 -2 -1 7 8. 1 5 6. 4 27 28 29 27 25 4.
Given that:a. n = 14 and Σx = 325.5, find x̅.b. n = 45 and y̅ = 23.6, find the value of Σy.c. Σz = 4598 and z̅ = 52.25, find the number of values in the set of data.d. Σxf = 86 and x̅
Given q̅ = 22 and Σ(q − 4) = 3672, find the number of values of q.
The area of land used for growing wheat in a region is denoted by Σw hectares. Find an expression in w for the total area in square kilometres, given that 1 hectare is equivalent to 10000m2.
a. Find the median and the mode for the values of x given in the following table.b. Give one positive and one negative aspect of using each of the median and the mode as the average value for x.c.
Nine values are 7, 13, 28, 36, 13, 29, 31,13 and x.a. Write down the name and the value of the measure of central tendency that can be found without knowing the value of x.b. If it is known that x is
Find the modal class for x and for y in the following tables. Ca 0- 4 14-20 3-6 8. 7-11 12-20 ss 66 80 134
Find the mean of x and of y given in the following tables.a.b. 18.0 18.5 19.0 19.5 20.0 8. 10 17 24
The lengths of 2500 bolts, xmm, are summarised by Σ(x − 40) = 875. Find the mean length of the bolts.
Speeds, measured in metres per second, are denoted by x. Find the constant k such that kx denotes the speeds in kilometres per hour.
A small company sells glass, which it cuts to size to fit into window frames. How could the company benefit from knowing the modal size of glass its customers purchase?
For the data given in the following table, it is given that q̅ = 8 5/9.Calculate the value of a. 8. 10 13 11
Six data values are coded by subtracting 13 from each. Five of the coded values are 9.3, 5.4, 3.9, 7.6 and 2.2, and the mean of the six data values is 17.6. Find the sixth coded value.
The wind speeds, x miles per hour (mph), were measured at a coastal location at midday on 40 consecutive days and are presented in the following table.Abel wishes to calculate an estimate of the mean
Four classes of continuous data are recorded as 1–7, 8–16,17–20 and 21–25. The class 1–7 has a frequency of 84 and there is no modal class. Find the total frequency of the other three
Calculate an estimate of the mean of x and of y given in the following tables.a.b. 0
The SD card slots on digital cameras are designed to accommodate a card of up to 24mm in width. Due to low sales figures, a manufacturer suspects that the machine used to cut the cards needs to be
Given that 15 values of x are such that Σ(3x − 2) = 528, find x̅ and find the value of b such that Σ(0.5x − b) = 138.
A teacher recorded the quiz marks of eight students as 11, 13, 15, 15, 17, 18, 19 and 20.They later realised that there was a typing error, so they changed the mark of 11 to 1.Investigate what effect
The following table gives the heights, to the nearest 5cm, of a group of people.Given that the modal class is 140–150cm, find the least possible value of p. Heights (cm) 120-135 140-150 155-160
Data about the times, in seconds, taken to run 100 metres by n adults are given in the following table.By first investigating the possible values of a and of b, find the largest possible value of n,
An examination was taken by 50 students. The 22 boys scored a mean of 71% and the girls scored a mean of 76%. Find the mean score of all the students.
Sixteen bank accounts have been accidentally under-credited by the following amounts, denoted by $x.To calculate x̅ manually, Fidel and Ramon code these figures using (x – 917) and (x – 920),
For 20 values of y, it is given that Σ(ax − b) = 400 and Σ(bx − a) = 545. Given also that x̅ = 6.25, find the value of a and of b.
The following table shows the lifetimes, to the nearest 10 days, of a certain brand of light bulb.a. Use upper class boundaries to represent the data in a cumulative frequency graph and estimate the
The following histogram illustrates the masses, m kilograms, of the 216 sales of hay that a farmer made to customers last year.a. Show that a calculated estimate of the mean is equal to the median.b.
Three classes of continuous data are given as 0–4, 4–10 and 10–18. The frequency densities of the classes 0–4 and 10–18 are in the ratio 4 : 3 and the total frequency of these two classes
The following graph illustrates the times taken by 112 people to complete a puzzle.a. Estimate the median time taken.b. The median is used to divide these people into two groups. Find the median time
For the data shown in the following table, x has a mean of 7.15.a. Find the mean value of y given in the following table.b. Find a calculated estimate of the mean value of z given in the following
The masses, m kilograms, of 148 objects are summarised in the following table.Construct a cumulative frequency polygon on graph paper, and use it to estimate the number of objects with masses that
The table below shows the number of books read last month by a group of children.a. If the mean number of books read is exactly 3.75, find the value of q.b. Find the greatest possible value of q
A company employs 12 drivers. Their mean monthly salary is $1950. A new driver is employed and the mean monthly salary falls by $8. Find the monthly salary of the new driver.
Throughout her career, an athlete has been timed in 120 of her 400-metre races. Her times, denoted by t seconds, were recorded on indoor tracks 45 times and are summarised by Σ(t − 60) = 83.7,
The midpoint of the line segment between A and B is at (5.2,–1.2).Find the coordinates of the midpoint after the following transformations have been applied to A and to B.a. T: Translation by the
It is claimed on the packaging of a brand of battery that they can run a standard kitchen clock continuously for ‘at least 150 days on average’. Tests are carried out to find the length of time,
An internet service provider wants to know how customers rate its services. A questionnaire asks customers to tick one of the following boxes.How might the company benefit from knowing each of the
The mean age of the 16 members of a karate club is 26 years and 3 months. One member leaves the club and the mean age of those remaining is 26 years. Find the age of the member who left the club.
All the interior angles of n triangular metal plates, denoted by y°, are measured.a. State the number of angles measured and write down the value of y.b. Hence, or otherwise, find the value of
Twenty people leaving a cinema are each asked, “How many times have you attended the cinema in the past year?” Their responses are:6, 2, 13, 1, 4, 8, 11, 3, 4, 16, 7, 20, 13, 5, 15, 3, 12, 9, 26
In a particular city there are 51 buildings of historical interest. The following table presents the ages of these buildings, given to the nearest 50 years.a. Write down the lower and upper boundary
The reaction times, t seconds, of 66 participants were measured in an experiment and presented below.Time (t seconds) ...................... No.
Jamila noted each student’s answer when her year group was asked to name their favourite colour.a. List the methods of representation that would be suitable for displaying Jamila’s data.b. Jamila
The weights of 220 sausages are summarised in the following table.i. State how many sausages weighed between 50g and 60g.ii. On graph paper, draw a histogram to represent the weights of the sausages.
The masses, m grams, of 690 medical samples are given in the following table.a. Find the value of p that appears in the table.b. On graph paper, draw a histogram to represent the data.c. Calculate an
The following table shows the widths of the 70 books in one section of a library, given to the nearest centimetre.a. Given that the upper boundary of the first class is 14.5cm, write down the upper
A large number of chickens’ eggs are individually weighed. The masses are grouped into nine classes, each of width 2 grams, from 48 to 66g. Name a type of representation in which the fact could be
The lengths of children’s feet, in centimetres, are classified as 14–16,17–19, 20–22 and so on. State the lower class boundary, the class width and the class mid-value for the lengths given
This stem-and-leaf diagram shows the number of employees at 20 companies. a. What is the most common number of employees?b. How many of the companies have fewer than 25 employees?c. What
The table below shows the heights, in metres, of 50 boys and of 50 girls.a. How many children are between 1.3 and 1.6 metres tall?b. Draw a histogram to represent the heights of all the boys and
Measurements of the distances, xmm, between two moving parts inside car engines were recorded and are summarised in the following table. There were 156 engines of type A and 156 engines of type B.a.
The heights of 600 saplings are shown in the following table.a. Suggest a suitable value for u, the upper boundary of the data.b. Illustrate the data in a histogram.c. Calculate an estimate of the
Over a 10-day period, Alina recorded the number of text messages she received each day. The following stem-and-leaf diagram shows her results.a. On how many days did she receive more than 10 but not
The runs scored by two batsmen in 15 cricket matches last season were:Batsman P: 53, 41, 57, 38, 41, 37, 59, 48, 52, 39, 47, 36, 37, 44, 59.Batsman Q: 56, 48, 31, 64, 21, 52, 45, 36, 57, 68, 77, 20,
The densities, d g/cm3, of 125 chemical compounds are given in the following table.Find the frequencies a, b, c and d given in the table below. Density (d glem') d
This back-to-back stem-and-leaf diagram shows the percentage scores of the 25 students who were the top performers in an examination.The 25 students are arranged in a line in the order of their
Boxes of floor tiles are to be offered for sale at a special price of $75. The boxes claim to contain at least 100 tiles each.a. Why would it be preferable to use a stem-and-leaf diagram rather than
The capacities of ten engines, in litres, are given rounded to 2 decimal places, as follows:1.86, 2.07, 1.74, 1.08, 1.99, 1.96, 1.83, 1.45, 1.28 and 2.19.These capacities are to be grouped in three
Over a 14-day period, data were collected on the number of passengers travelling on two ferries, A and Z. The results are presented to the right.a. How many more passengers travelled on ferry Z than
The diameters, d cm, of 60 cylindrical electronic components are represented in the following cumulative frequency graph.a. Find the number of components such that 0.2 ≤ d < 0.4, and explain why
A charity group’s target is to raise a certain amount of money in a year. At the end of the first month the group raised 36% of the target amount, and at the end of each subsequent month they
Each of the 70 trainees at a secretarial college was asked to type a copy of a particular document. The times taken are shown, correct to the nearest 0.1 minutes, in the following
The following cumulative frequency graph shows the masses, m grams, of 152 uncut diamonds.a. Estimate the number of uncut diamonds with masses such that:i. 9 ≤ m < 17ii. 7.2 ≤ m < 15.6.b.
University students measured the heights of the 54 trees in the grounds of a primary school. As part of a talk on conservation at a school assembly, the students have decided to present their data
The following stem-and-leaf diagram shows eight randomly selected numbers between 2 and 4.Given that a – b = 7 and that the sum of the eight numbers correct to the nearest integer is 24, find the
The total numbers of eggs laid in the nests of two species of bird were recorded over several breeding seasons.The numbers of eggs laid in the nests of 10 wrens and 10 dunnocks are:Wrens: 22, 18, 21,
A railway line monitored 15% of its August train journeys to find their departure delay times. The results are shown below. It is given that 24 of these journeys were delayed by less than 2
Eighty people downloaded a particular application and recorded the time taken for the download to complete. The times are given in the following table.a. Find the number of downloads that completed
A university investigated how much space on its computers’ hard drives is used for data storage. The results are shown below. It is given that 40 hard drives use less than 20GB for data storage.a.
The daily journey times for 80 bank staff to get to work are given in the following table.a. How many staff take between 15 and 45 minutes to get to work?b. Find the exact number of staff who take x+
Last year Tom renovated an old building during which he worked for at least 9 hours each week. By plotting four points in a graph, he has represented the time he spent working.a. What can you say
A histogram is drawn with three columns whose widths are in the ratio 1: 2 : 4. The frequency densities of these classes are in the ratio 16 :12 : 3, respectively.a. Given that the total frequency of
The lengths of the 575 items in a candle maker’s workshop are represented in the histogram.a. What proportion of the items are less than 25cm long?b. Estimate the number of items that are between
A fashion company selected 100 12-year-old boys and 100 12-year-old girls to audition as models. The heights, hcm, of the selected children are represented in the following graph.a. What features of
The following table shows the focal lengths, l mm, of the 84 zoom lenses sold by a shop. For example, there are 18 zoom lenses that can be set to any focal length between 24 and 50mm.a. What feature
The histograms below illustrate the number of hours of sunshine during August in two regions, A and B. Neither region had more than 8 hours of sunshine per day.a. Explain how you know that some
The thicknesses, kmm, of some steel sheets are represented in the histogram. It is given that k < 0.4 for 180 sheets.a. Find the ratio between the frequencies of the three classes. Give your
The following table shows the ages of the students currently at a university, given by percentage. Ages are rounded down to the number of whole years.a. Represent the data in a percentage cumulative
The following table shows estimates, in hundred thousands, of the number of people living in poverty and the populations, in millions, of the countries where they live.Represent, in a single diagram,
A hotel has 90 rooms. The table summarises information about the number of rooms occupied each day for a period of 200 days.i. Draw a cumulative frequency graph on graph paper to illustrate this
The masses, in kilograms, of the animals treated at a veterinary clinic in the past year are illustrated in a histogram. The histogram has four columns of equal height. The following table shows the
The distances, in km, that 80 new cars can travel on 1 litre of fuel are shown in the table.These distances are 10% greater than the distances the cars will be able to travel after they have covered
The minimum daily temperature at a mountain village was recorded to the nearest 0.5°C on 200 consecutive days. The results are grouped into a frequency table and a histogram is drawn.The
A small company produces cylindrical wooden pegs for making garden chairs. The lengths and diameters of the 242 pegs produced yesterday have been measured independently by two employees, and their
The percentage examination scores of 747 students are given in the following table.Given that the frequency densities of the four classes of percentage scores are in the ratio 5 : 8 :13: 7, find the
Find the mode(s) of the following sets of numbers.a. 12,15,11, 7, 4,10, 32,14, 6,13,19, 3 b. 19, 21, 23,16, 35, 8, 21,16,13,17,12,19,14, 9
Calculate the mean of the following sets of numbers:a. 28,16, 83, 72,105, 55, 6 and 35b. 7.3, 8.6, 11.7, 9.1, 1.7 and 4.2c. 3 ½, 5 ¼, 9 ¾, 4 ¼, and 7 3/8,

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