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mathematics
cambridge international as & a level mathematics probability & statistics
Cambridge International AS & A Level Mathematics Probability & Statistics 1 Coursebook 1st Edition Dean Chalmers, Julian Gilbey - Solutions
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 milligramsb. Kilograms.
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 = 7 1/6 = , find the value of Σf .e. Σf = 135 and x̅ = 0.842, find the value of Σxf .
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. Some values in the table have been incorrectly recorded as 8 instead of 4. Find the number of
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 greater than 40, which other measure of central tendency can be found and what is its value?c. If
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 wind speed in kilometres per hour (km/h). He knows that a distance of 5miles is approximately equal
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 classes.
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 recalibrated. The widths, wmm, of 400 of these cards were measured and are coded in the following
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 this change has on the mode, mean and median of the students’ marks.
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) 120-135 140-150 155-160 165–170 175-185 No. people 30 12 16 21 P
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, given that the modal class contains the slowest runners. Time (xs) 13.6sx
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), respectively.Who has the simpler maths to do? Explain your answer. 917.95 917.98 918.03 917.97 918.01
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 median lifetime of the light bulbs.b. How might the manufacturer choose a value to use as 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. Estimate the price per kilogram at which the hay was sold, given that these sales generated exactly
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 is 120. Find the least possible frequency of the modal class, given that the modal class is 4–10.
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 taken by each of the groups. 120 - 80 - 40 - 6 10 Time (min) Cumulative frequency (No. peo ple) 2.
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 table. 3 6 10 15 Frequency b. d a
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 are:a. Within 0.1kg of the medianb. More than 200g from the median. Mass (m kg) m
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 if:i. The modal number of books read is 4.ii. The median number of books read is 4. No. bock 4 No.
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, and on outdoor tracks where Σ(t − 65) = −38.7. Calculate her average 400-metre running time
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 vectorb. E: Enlargement through the origin with scale factor 5.c. Transformations T and E are carried
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, t hours, that a standard kitchen clock runs using one of these batteries. The results are shown in
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 available average responses of its customers?b. What additional benefit could the company obtain by
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. Give a reason why your answer might not be very accurate.
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 Σ(y − 30).
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 and 10.Construct a stem-and-leaf diagram for these data and include a key.
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 values of the class containing the greatest number of buildings.b. State the widths of the four
The reaction times, t seconds, of 66 participants were measured in an experiment and presented below.Time (t seconds) ...................... No. participants (cf )t < 1.5 ................................................................... 0t <
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 wishes to emphasise that the favourite colour of exactly three-quarters of the students is blue.
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. Weight erams) < 20 < 30
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 estimate of the number of samples with masses between 8 and 18 grams. Mass (m grams) 4sm
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 boundary of the second class.b. Draw up a cumulative frequency table for the data and construct a
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 seen that the majority of the eggs have masses from 54 to 60g. Explain how the representation would
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 as 17–19.
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 percentage of the companies have more than 30 employees?d. Determine which of the three rows in the
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 girls together.c. Estimate the number of children whose heights are 1.7 metres or more. Height (m) 1.2-
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. Draw and label two cumulative frequency curves on the same axes.b. Use your graphs to estimate:i.
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 number of saplings with heights that are:i. Less than 25cm ii. Between 7.5 and 19.5cm. Height
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 more than 15 messages?b. How many more rows would need to be added to the stem-and-leaf diagram if
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, 42, 51, 71.a. Show the data in a diagram that allows easy comparison of the two performances.b.
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 scores. Describe the student in the middle of the line and find the greatest possible number of boys in
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 a bar chart to represent the numbers of tiles, which are 112, 116, 107, 108, 121, 104, 111, 106, 105
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 classes as 1.0–1.4, 1.5–2.0 and 2.1–2.2.a. Find the frequency of the class 1.5–2.0 litres.b.
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 on ferry A?b. The cost of a trip on ferry A is $12.50 and the cost of a trip on ferry Z is $x. The
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 your answer is not an estimate.b. Estimate the number of components that have:i. A diameter of less
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 manage to raise exactly half of the amount outstanding.a. How many months will it take the group to raise
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 table.a. Explain why the interval for the first class has a width of 0.3 minutes.b. Represent the times taken in a
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. The lightest 40 diamonds are classified as small. The heaviest 40 diamonds are classified as large.
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 using one of the following diagrams.a. Give one disadvantage of using each of the representations
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 value of a and of b. 2|133 a Key: 2 |1 3 b 5 67 represents 2.1
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, 23, 17, 23, 20, 19, 24, 13.Dunnocks: 28, 24, 23, 19, 30, 27, 22, 25, 22, 17.a. Represent the data
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 minutes.a. How many journeys were monitored?b. Calculate an estimate of the number of these journeys that
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 in 5 to 6 minutes.b. On a histogram, the download times from 5 to 6 minutes are represented by a
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. Find the total number of hard drives represented.b. Calculate an estimate of the number of hard
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+ y/2 minutes or more to get to work, given that 85% of the staff take less than x minutes and that
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 about the time that Tom spent working on the basis of this graph?b. Explain why Tom’s graph might be
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 data is 390, find the frequency of each class. [3]b. The classes with the two highest
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 12.4 and 36.8cm long.c. The shortest 20% of the workshop’s items are to be recycled. Calculate an
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 data suggest that the children were not selected at random?b. Estimate the number of girls who
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 of the data does not allow them to be displayed in a histogram?b. What type of diagram could you use
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 information for one of the regions has been omitted.b. After studying the histograms, two students make
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 answer in simplified form.b. Find the value of n, given that frequency density measures sheets per nmm.c.
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 frequency polygon.b. The oldest 8% of these students qualify as ‘mature’. Use your polygon to
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, the actual numbers and the relative poverty that exists in these countries. In what way do the
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 information.ii. Estimate the number of days when more than 30 rooms were occupied.iii. On 75% of the
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 class intervals and the number of animals in two of the classes.a. Find the value of a and of b, and
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 more than 100000km.Estimate how many of the cars can travel 10.5km or more on 1 litre of fuel when
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 temperatures ranged from 0.5°C to 2°C on n days, and this class is represented by a column of height hcm.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 results are given in the following table.a. On the same axes, draw two cumulative frequency graphs:
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 value of p and of q. Score (%) P-50 51-70 71-80 pria 81-9 165 240 195 147
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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