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Cambridge International AS & A Level Mathematics: Probability & Statistics 1 Practice Book
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Cambridge InternationalAS & A Level Mathematics:
Probability &
Statistics 1
Practice Book
,Contents
How to use this book
1 Representation of data
2 Measures of central tendency
3 Measures of variation
4 Probability
5 Permutations and combinations
6 Probability distributions
7 The binomial and geometric distributions
8 The normal distribution
The standard normal distribution function
Answers
, How to use this book
Throughout this book you will notice particular features that are designed to help your learning.
This section provides a brief overview of these features.
■ Display numerical data in stem-and-leaf diagrams, histograms and cumulative frequency
graphs.
■ Interpret statistical data presented in various forms.
■ Select an appropriate method for displaying data.
Learning objectives indicate the important concepts within each chapter and help you to navigate
through the practice book.
TIP
On the evening when 30 people viewed films on screen A, there could have been as few as 37
or as many as 79 people viewing films on screen B.
Tip boxes contain helpful guidance about calculating or checking your answers.
WORKED EXAMPLE 2.1
The mass, x kg, of the contents of 250 bags of bird seed are recorded in the following table.
Mass (x kg) 2.48 ⩽ x < 2.49 2.49 ⩽ x < 2.51 2.51 ⩽ x < 2.56 2.56 ⩽ x < u
No. bags (f) 19 48 98 85
Given that the modal class is 2.49 ⩽ x < 2.51, find to 2 decimal places the least possible
value of u.
Answer
For the modal class, frequency density
For 2.56 ⩽ x < u, frequency density
The least possible value of u is 2.92
Worked examples provide step-by-step approaches to answering questions. The left side shows a fully
worked solution, while the right side contains a commentary explaining each step in the working.
E
Extension material goes beyond the syllabus. It is highlighted by a red line to the left of the text.
, END-OF-CHAPTER REVIEW EXERCISE 1
1 A set of electronic weighing scales gives masses in grams correct to 3 decimal places. Jan
has recorded the masses, m grams, of a large number of small objects, and he finds that
5.020 ⩽ m < 5.080 for 80 of them. Jan decides to illustrate the data for these 80 objects in a
stem-and-leaf diagram.
a List an appropriate set of numbers that Jan can write into the stem of his diagram.
b Write down the least possible mass of any one of these 80 objects.
The End-of-chapter review exercise contains exam-style questions covering all topics in the chapter.
You can use this to check your understanding of the topics you have covered.
Throughout each chapter there are exercises containing practice questions. The questions are coded:
PS These questions focus on problem solving.
P These questions focus on proofs.
M These questions focus on modelling.
You should not use a calculator for these questions.
You can use a calculator for these questions.
,Chapter 1
Representation of data
■ Display numerical data in stem-and-leaf diagrams, histograms and cumulative frequency
graphs.
■ Interpret statistical data presented in various forms.
■ Select an appropriate method for displaying data.
,1.1 Types of data and 1.2 Representation of discrete data: stem-and-leaf
diagrams
WORKED EXAMPLE 1.1
Correct to the nearest centimetre, the length of each of the 80 pencils in a box is 18 cm.
a State the lower boundary and the upper boundary of the length of a pencil.
b What is the least possible total length of all the 80 pencils together?
c The 80 pencils are shared by six children so that each receives an odd number, and
no two children receive the same number of pencils. Draw an ordered stem-and-leaf
diagram showing one of the possibilities for the number of pencils given to the
children.
Answer
a Lower boundary = 17.5 cm Lengths rounded to 18 cm mean 17.5 ⩽
Upper boundary = 18.5 cm length < 18.5 cm.
b 80 × 17.5 = 1400 cm
c 0 1 5 9 Key: 1|3 The diagram must show six different,
ordered odd numbers with a sum of 80.
1 3 represents 13 One possible solution, shown here, is to
use 1, 5, 9, 13, 23 and 29.
2 3 9 pencils
EXERCISE 1A
1 Sara has collected three sets of data from the children in her daughter’s class at school.
These are: A: their first names; B: their heights; C: their shoe sizes. Match each set of
data with the one word from the following list that best describes it.
X: discrete Y: qualitative Z: continuous
2 a Correct to the nearest ten metres, the perimeter of a rectangular football pitch is
260 metres. Complete the following inequality which shows the lower and upper
boundaries of the actual perimeter:
………… m ⩽ perimeter < ………… m
b Eliana has 16 coins. The mass of each coin, correct to 1 decimal place, is 2.4 grams.
Find the least possible total mass of the 16 coins.
3 A car was driven a distance of 364km in five hours. The distance driven is correct to the
nearest kilometre and the time taken is correct to the nearest hour.
Find the lower boundary and the upper boundary of the average speed of the car.
4 The numbers of items purchased by the first 11 customers at a shop this morning were
6, 2, 13, 5, 1, 7, 2, 11, 16, 20 and 15.
a Display these data in a stem-and-leaf diagram and include an appropriate key.
b Find the number of items purchased by the 12th customer, given that the first 12
customers at the shop purchased a total of 111 items.
5 a Correct to the nearest metre, the length and diagonal of a rectangular basketball
court measure 18m and 23m, respectively. Calculate the lower boundary of the width
of the court, correct to the nearest 10cm.
b Correct to the nearest 0.01 cm, the radius of a circular coin is 0.94 cm. Find the least
number of complete revolutions that the coin must be turned through, so that a point
on its circumference travels a distance of at least 9.5 metres.
6 Bobby counts the number of grapes in 12 bunches that are for sale in a shop.
To illustrate the data, he first produces the following diagram.
, 2 8 5 9
3 4 7 5 4
4 2 1 3 3 5
a State whether the data are:
i qualitative or quantitative
ii discrete or continuous.
b Complete Bobby’s work by ordering the data in a stem-and-leaf diagram and adding
a key.
7 Construct stem-and-leaf diagrams for the following data sets.
a The speeds, in kilometres per hour, of 20 cars, measured on a city street:
41, 15, 4, 27, 21, 32, 43, 37, 18, 25, 29, 34, 28, 30, 25, 52, 12,
36, 6, 25
b The times taken, in hours (to the nearest tenth), to carry out repairs to 17 pieces of
machinery:
0.9, 1.0, 2.1, 4.2, 0.7, 1.1, 0.9, 1.8, 0.9, 1.2, 2.3, 1.6, 2.1, 0.3,
0.8, 2.7, 0.4
M 8 The contents of 30 medium-size packets of soap powder were weighed and the results,
in kilograms correct to 4 significant figures, were as follows.
1.347 1.351 1.344 1.362 1.338 1.341 1.342 1.356 1.339 1.351
1.354 1.336 1.345 1.350 1.353 1.347 1.342 1.353 1.329 1.346
1.332 1.348 1.342 1.353 1.341 1.322 1.354 1.347 1.349 1.370
a Construct a stem-and-leaf diagram for the data.
b Why would there be no point in drawing a stem-and-leaf diagram for the data
rounded to 3 significant figures?
PS 9 Two films are shown on screen A and screen B at a cinema each evening. The numbers
of people viewing the films on 12 consecutive evenings are shown in the back-to-back
stem-and-leaf diagram.
Screen A (12) Screen B (12)
0 3 7
8 3 4
7 6 4 0 5 3 4
7 4 1 6 4 5 6 7 8
9 2 7 1 6 8 9
Key: 1|6|4 represents 61 viewers for A and 64 viewers for B
A second stem-and-leaf diagram (with rows of the same width as the previous diagram)
is drawn showing the total number of people viewing films at the cinema on each of
these 12 evenings. Find the least and greatest possible number of rows that this second
diagram could have.
TIP
On the evening when 30 people viewed films on screen A, there could have been as few
as 37 or as many as 79 people viewing films on screen B.
10 The masses, to the nearest 0.1 g, of 30 Yellow-rumped and 30 Red-fronted Tinkerbirds
were recorded by Biology students. Their results are given in the tables below:
Yellow-rumped
17.0 13.1 15.7 14.7 17.4 15.3
14.2 16.2 16.9 16.8 16.5 14.2
15.5 16.9 13.5 18.1 17.6 17.9
13.0 15.1 18.2 17.8 18.1 12.3
12.2 17.7 16.5 16.7 14.8 16.6
Red-fronted
18.2 14.7 15.9 14.7 15.2 14.5
, 17.3 13.2 14.0 20.2 17.5 15.6
19.3 19.4 17.4 16.5 16.8 15.8
15.3 13.3 16.0 18.6 13.6 18.0
14.1 15.5 20.0 13.9 16.8 14.7
a Display the masses in a back-to-back stem-and-leaf diagram.
b How many Yellow-rumped Tinkerbirds are heavier than the lightest 80% of the Red-
fronted Tinkerbirds?
c The students decide to display the birds’ masses, correct to the nearest 5 grams, in
two bar charts.
i Write down the frequencies for the three classes of Yellow-rumped Tinkerbirds.
ii Explain why it is possible for the 20 g class of Red-fronted Tinkerbirds to have
the same frequency as the 20 g class of Yellow-rumped Tinkerbirds.
iii Given that there are equal numbers of Tinkerbirds in the two 20 g classes,
construct two bar charts on the same axes showing the masses to the nearest 5 g.
,1.3 Representation of continuous data: histograms
WORKED EXAMPLE 1.2
The percentage marks scored by 100 candidates in an examination are shown in equal-width
intervals in the following histogram.
The marks are to be regrouped in four unequal-width intervals, as shown in the table.
Marks (%) f Frequency density
9.5 ⩽ x < 19.5 6 6 ÷ 10 = 0.6
19.5 ⩽ x < 39.5 a
39.5 ⩽ x < 69.5 b
69.5 ⩽ x < 89.5 c
a Find the frequencies a, b and c.
b By calculating the three missing frequency densities, illustrate these data in a
histogram with four unequal-width intervals.
Answer
a
Column area ∝ class
Marks (%) Column No. candidates
frequency, and we know that
area (f)
the sum of the frequencies is
9.5 ⩽ x < 10 × 0.6 = 6 6 100. This allows us to draw
19.5 up a grouped frequency table,
which corresponds with the
19.5 ⩽ x < 10 × 0.8 = 8 8 original histogram.
29.5
29.5 ⩽ x < 10 × 1.0 = 10
39.5 10
39.5 ⩽ x < 10 × 1.3 = 13
49.5 13
49.5 ⩽ x < 10 × 1.5 = 15
59.5 15
59.5 ⩽ x < 10 × 2.3 = 23
69.5 23
69.5 ⩽ x < 10 × 1.4 = 14
79.5 14
79.5 ⩽ x < 10 × 1.1 = 11
89.5 11
Total area = Σf = 100
, 100
Add together the relevant
frequencies.
b
The required histogram
showing the marks in four
unequal-width intervals is
shown.
EXERCISE 1B
1 The speeds, in km h−1, of 200 vehicles travelling on a highway were measured by a
radar device. The results are summarised in the following table. Draw a histogram to
illustrate the data.
Speed 45− 60− 75− 90− 105− 120 or more
Frequency 12 32 56 72 20 8
2 The mass of each of 60 pebbles collected from a beach was measured. The results,
correct to the nearest gram, are summarised in the following table. Draw a histogram of
the data.
Mass 5−9 10−14 15−19 20−24 25−29 30−34 35−44
Frequency 2 5 8 14 17 11 3
3 Thirty calls made by a telephone saleswoman were monitored. The lengths in minutes,
to the nearest minute, are summarised in the following table.
Length of call 0−2 3−5 6−8 9−11 12−15
No. calls 17 6 4 2 1
a State the boundaries of the first two classes.
b Illustrate the data with a histogram.
4 The haemoglobin levels in the blood of 45 hospital patients were measured. The results,
correct to 1 decimal place and ordered for convenience, are as follows.
9.1 10.1 10.7 10.7 10.9 11.3 11.3 11.4 11.4 11.4 11.6 11.8 12.0
12.1 12.3
12.4 12.7 12.9 13.1 13.2 13.4 13.5 13.5 13.6 13.7 13.8 13.8 14.0 14.2
14.2
14.2 14.6 14.6 14.8 14.8 15.0 15.0 15.0 15.1 15.4 15.6 15.7 16.2 16.3
16.9
a Form a grouped frequency table with eight classes.
b Draw a histogram of the data.
5 The table shows the age distribution, in whole numbers of years, of the 200 members of
a chess club.
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