Exercise – Unit 6
Preparation for HKDSE (Junior Topics)
Preparation for HKDSE (Junior Topics) Exercise
Unit 6 Statistics and Basic Probability
NOTES
1.
Statistics
The arithmetic mean, the median and the mode/modal class are three different measures
of the central tendency of a group of data.
Measure of
central
tendency
Ungrouped data
(for a set of data x1, x2, …, xN)
Arithmetic
sum of data
Mean 
number of data
mean
x  x2  ...  xN
 1
N
Median Suppose the data x1, x2, …, xN are arranged in
ascending order.
When N is odd, median
 N  1
 the 
 th datum
 2 
When N is even, median

1 N
N

  the   th datum  the   1 th datum 
2  2 
2


Mode /
Modal
class
2.
(a)
The mode is the datum with the highest
frequency.
Grouped data
(for a set of data with class marks
x1, x2, …, xN and corresponding
frequencies f1, f2, …, fN)
Mean 
f1 x1  f 2 x2  ...  f N xN
f1  f 2  ...  f N
The median can be found from
the cumulative frequency
polygon (or curve).
The modal class is the class
interval with the highest
frequency.
Basic Probability
In the case of an activity where all the possible outcomes are equally likely, the
probability of an event E, denoted by P(E), is given by
number of outcomes favourable to the event E
P( E ) 
,
total number of possible outcomes
where 0  P( E )  1 .
1
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Exercise – Unit 6
(b)
Preparation for HKDSE (Junior Topics)
Expected Value
Consider an activity with n possible outcomes, and the values obtained from
the possible outcomes are x1, x2, x3, …, xn. If the probabilities of occurrence of
the above possible outcomes are p1, p2, p3, …, pn respectively, then the
expected value of this activity is x1p1 + x2p2 + x3p3 + … + xnpn.
STRUCTURAL QUESTIONS
Section A(1)
1. The pie chart on the right shows the favourite basketball teams of a
group of teenagers. It is given that the number of teenagers whose
favourite basketball team is Eagle is 50% greater than the number of
teenagers whose favourite basketball team is Fire.
(a) Find x.
(b) What percentage of teenagers in the group whose favourite
basketball team is Cats?
2.
The table below shows the distribution of the numbers of siblings of a group of S1 students.
Number of siblings
0
1
2
3
Number of students
9
22
6
3
Find the mean, the median and the mode of the numbers of siblings of the group of S1 students.
3.
The table below shows the numbers of days late to school of some students in a month.
Number of days late to school
0
1
2
3
4
Number of students
25
8
x
6
5
If the mean number of days late to school of these students is 1.16, find the value of x.
4.
The bar chart on the right shows the numbers of
handbags owned by a group of girls.
(a) If the median of the numbers of handbags
owned by the group of girls is 1.5, find the
value of k.
(b) A girl leaves this group. It is known that the
number of handbags owned by this girl is 2.
Find the change in the median of the numbers
of handbags owned by the group of girls due to
the leaving of this girl.
2
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Exercise – Unit 6
5.
Preparation for HKDSE (Junior Topics)
The following table shows the distribution of the numbers of working hours of a group of
staff in a company on a certain day.
Number of working hours
7
8
9
10
Number of staff
2
x
5
1
It is given that x is a positive integer.
(a) If the median of the numbers of working hours of the group of staff is 9, find the
possible value(s) of x.
(b) On that day, two new staff joined the company. They both worked 8 hours on that day.
After adding their data to the above distribution, the median of the numbers of working
hours of the staff remains unchanged. Find the possible value(s) of x.
6.
The following stem-and-leaf diagram shows the scores of a group of students in a
Mathematics quiz.
Stem (10 marks)
Leaf (1 mark)
0
1
2
3
6
4
0
0
7
4
2
0
9
6
3
0
7
8
3
9
(a) Find the mean and the median of the above distribution.
(b) A student leaves this group. The marks obtained by this student is 28 marks. Find the
changes in the mean and the median due to the leaving of this student.
7.
Jason chooses a letter from each of the two words ‘JUNE’ and ‘REST’. Find the probability
that only one of the letters chosen is a vowel.
8.
Box A contains 1 yellow ball, 2 green balls and 1 red ball, while box B contains 2 yellow balls
and 2 red balls. If a ball is randomly drawn from each box, find the probability that both of
them are yellow.
9
Two fair dice are thrown. Find the probability that the sum of the numbers obtained is a prime
number.
3
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Exercise – Unit 6
Preparation for HKDSE (Junior Topics)
Section A(2)
10. The stem-and-leaf diagram below shows the distribution of the numbers of training hours of
the members of a basketball team in a month.
Stem (10 hours)
1
2
3
Leaf (1 hour)
3
4
4
7
7
8
8
1
1
1
4
4
(a) Find the mean, the median and the mode of the above distribution.
(b) Two new members joined the team and they attended more than 24 hours of training in
that month. After adding their data to the above distribution, the mean of the distribution
is increased by 1 hour while both the median and the mode of the distribution remain
unchanged. Find the training time of each of the new member.
11. The chart below shows the distribution of the blood type of a group of students.
If a student is randomly selected from the group, then the probability that the student is of
blood type B is 0.3.
(a) Find k.
(b) Suppose that the above distribution represented by a pie chart.
(i) Find the angle of the sector representing that the group of students of blood type AB.
(ii) Some students who are of blood type O were mistakenly recorded as type AB blood.
If the data are corrected, will the angle of the sector representing the group of
students of blood type AB is equal to that representing the group of students of
blood type O? Explain your answer.
4
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Exercise – Unit 6
Preparation for HKDSE (Junior Topics)
12. The scores (in marks) of the students of class A in a language test are shown as follows:
42
44
45
45
50
53
53
53
60
60
64
64
68
70
(a) Write down the median and the mode of the scores of the students of class A in the
language test.
(b) The stem-and-leaf diagram below shows the scores (in marks) of the students of class B
in a Mathematics test. It is given that the mode of the distribution is 58.
Stem (10 marks)
Leaf (1 mark)
4
8
9
5
6
a
b
8
8
6
0
0
2
9
9
(i) Find a and b.
(ii) From each class, a student is randomly selected. If the sum of their scores is greater
than 119 marks, then they can join a competition. Find the probability that the two
students can join the competition.
13. The stem-and-leaf diagram below shows the amount of money spent on transportation in a
week by 20 students.
Stem ($10)
Leaf ($1)
0
3 7
8
1
0 0
0
0
1
2
3
4
4
6
2
3
1
3
2
5
3
6
8
(a) Find the mean and the mode of the amounts of money spent on transportation by the 20
students.
(b) The data of the amounts of money spent on transportation in that week by other 5 students
are collected. It is known that the mean of these 5 data is $12.8, and two of the data are
$16 and $21.
(i)
Find the mean of the amounts of money spent on transportation in that week by
these 25 students.
(ii) Aaron claims that the mode of the amounts of money spent on transportation in that
week by these 25 students is the same as the mode found in (a). Do you agree?
Explain your answer.
5
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Exercise – Unit 6
5
Preparation for HKDSE (Junior Topics)
MULTIPLE CHOICE QUESTIONS
Section A
1. The pie chart shows the distribution of the time that Jack spent on
various activities in a certain week. If Jack spent 4 hours on
shopping in that week, find the total number of hours that he
spent on these activities in that week.
A.
B.
C.
D.
2.
32
33
45
48
The figure below shows a cumulative frequency curve.
Which of the following can be its corresponding histogram?
A.
B.
C.
D.
6
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Exercise – Unit 6
3.
4.
If y decreases when x increases, which of the following scatter diagrams may represent the
relation between x and y?
A.
B.
C.
D.
If the mode of the seven numbers 7, 4, 5, 7, x, y and z is 5, then the median of these seven
numbers is
A.
B.
C.
D.
5.
4.
5.
6.
7.
In a company, the mean age of 10 male staff and 6 female staff is 30.5. If the mean age of the
male staff is 35.3, then the mean age of the female staff is
A.
B.
C.
D.
6.
Preparation for HKDSE (Junior Topics)
22.5.
25.7.
27.6.
28.2.
If the mean of the five numbers a, b, c, d and e is 12.3, then the mean of the five numbers
18 – a, 18 – b, 18 – c, 18 – d and 18 – e is
A.
B.
C.
D.
5.7.
8.7.
12.3.
18.
7
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Exercise – Unit 6
7.
Preparation for HKDSE (Junior Topics)
Consider the following integers:
1
1
2
3
4
5
5 5
7
9
9
k
Let x, y and z be the mean, the median and the mode of the above integers respectively. If
2  k  4 , which of the following must be true?
x y
I.
II. x  z
III. y  z
A.
B.
C.
D.
8.
9.
I and II only
I and III only
II and III only
I, II and III
Consider the following data:
16 18 9
10 12 12 13 13 x
y
If both the mean and the median of the data are 12, which of the following must be true?
I.
x  13
II. y  12
III. x  y  17
A.
B.
C.
I only
II only
I and III only
D.
II and III only
It is given that x1 ≤ x2 ≤ x3 ≤ … ≤ x20. Let p1, q1 and r1 be the mean, the median and the mode
of a group of numbers {x1, x2, x3, … , x20} respectively. If p2, q2 and r2 are the mean, the
median and the mode of a group of numbers {x1, x2, x3, … , x20, q1} respectively, which of
the following must be true?
I. p1 = p2
II. q1 = q2
III. r1 = r2
A.
I only
B.
C.
D.
II only
I and III only
II and III only
8
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Exercise – Unit 6
Preparation for HKDSE (Junior Topics)
10. The bar chart on the right shows the distribution of the numbers of books read by a class of
students in a certain week. If a student is selected randomly from the class, find the
probability that the selected student read 3 books in that week.
1
A.
8
1
B.
4
3
C.
16
5
D.
16
11. A fair dice is thrown twice. Find the probability that the product of the numbers obtained is
an even number.
1
A.
4
1
B.
3
1
C.
2
3
D.
4
12. An urn contains x white balls, 4 black balls and 8 red balls. If a ball is randomly drawn from
1
the urn, the probability of selecting a black ball is
. Find the value of x.
x
A. 3
B. 4
C. 6
D. 12
13. 3 is a 3-digit number, where  is an integer from 0 to 8 inclusively and  is an integer
from 0 to 7 inclusively. Find the probability that the 3-digit number is divisible by 5.
1
A.
8
3
B.
14
1
C.
4
1
D.
3
9
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Exercise – Unit 6
Preparation for HKDSE (Junior Topics)
14. Two fair dice are thrown in a game. If the sum of the two numbers is greater than 9, $54 will
be gained; otherwise, $18 will be gained. Find the expected gain of the game.
A.
B.
C.
D.
$23
$24
$28
$29
15. The stem-and-leaf diagram below shows the distribution of the ages of teachers in a school.
Stem (tens)
Leaf (units)
2
4
9
9
3
4
0
1
3
4
4
7
7
8
9
5
2
3
3
4
6
7
8
8
A teacher is randomly selected from the school. Find the probability that the selected teacher
is above 45 years old.
A.
B.
C.
D.
0.4
0.5
0.6
0.7
10
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