Download TOPIC#01 Statistics – Bar Charts, Frequency Polygons, Histograms

TOPIC#01
Statistics – Bar Charts, Frequency Polygons, Histograms, Means,
Median & Mode
Q1.
A survey was made of the number of people in each car that crossed a bridge one morning.
The table shows the results.
Number of people in the car
1
2
3
4
Number of cars
x
8
2
3
(a)
Write down the largest possible value of x given that the mode is 2.
(b)
Write down the largest possible value of x given that the median is 2.
(c)
Calculate the value of x given that the mean is 2.
Ans.
(a)
7
Q2.
In a school of 120 children, a teacher found out the distance, d meters, that each child could
(b)
12
(c)
8
swim. The results were grouped as shown in the table.
(a)
Group A
Group B
Group C
Distance (d meters)
0 ≤ d < 100
100 ≤ d < 200
200 ≤ d < 400
Number of children
30
50
40
A pie chart is drawn to represent this information. Calculate the angle which
represents the number of children in Group B.
(b)
A histogram is drawn to represent the information. Calculate the frequency
densities for each group.
150°
(b)
3
1
1
, Group B : , Group C :
10
2
5
Ans.
(a)
Q3.
Eleven people work for a company. Five office workers each have a car allowance of $10,000.
Group A :
Six managers have car allowance of.
$12,000, $18,000, $20,000, $24,000, $28,000 and $35,000.
(a)
State
(i)
the median car allowance,
(ii)
the mode of this distribution.
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TOPIC#01
Statistics – Bar Charts, Frequency Polygons, Histograms, Means,
Median & Mode
(b)
Calculate the mean car allowance.
(c)
Two people are chosen to random from the eleven members of the company.
Calculate the probability that one is a manager and one an office worker. Express
your answer as a fraction.
(i)
$21 000
(ii)
$10 000
(b)
#17 000
(c)
6
11
Ans.
(a)
Q4.
Over a 40 day period, the number of students absent from school was recorded. The results
are given in the table below.
Number of students
0-4
5-9
10-14
15-19
20-29
Number of days
3
8
10
13
6
For example, on 8 of the days there were 5, 6, 7, 8 or 9 students absent from school.
(a)
Which is the modal class?
(b)
Calculate an estimate of the mean number of students absent per day.
Ans.
(a)
15 – 19
Q5.
Mr. Smith asked the children in his class ‘What is your favourite colour? their replies are
(b)
13.75
given below:
(a)
Green
Blue
Green
Yellow
Blue
Green
Red
Blue
Green
Blue
Yellow
Green
Yellow
Blue
Yellow
Blue
Blue
Green
Blue
Yellow
Green
Blue
Green
Yellow
Blue
By making tally marks, or otherwise, obtain the frequency distribution of the
colours.
Colour
Frequency
Green
Blue
Red
Yellow
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TOPIC#01
Statistics – Bar Charts, Frequency Polygons, Histograms, Means,
Median & Mode
(b)
Ans.
(a)
(b)
Q6.
State the mode of this distribution.
Colour
Frequency
Green
8
Blue
10
Red
1
Yellow
6
Blue
In 2000 Esther went to a tennis tournament. Her ticket cost $35. At the tournament she
bought a programme costing $3 and an ice cream costing $2.
(a)
This information is to be shown on a pie chart. Calculate the angle of the sector
which represents the amount she spent on ice cream.
(b)
In 2001 the cost of a ticket was $36.75. Calculate the percentage increase in the cost
of a ticket.
Ans.
(a)
18°
(b)
5%
Q7.
(a)
An article in a newspaper reported that the number of crimes had been reduced by
half from 1991 to 2001. The article contained that bar chart shown here. Explain why
this bar chart might be considered misleading.
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TOPIC#01
Statistics – Bar Charts, Frequency Polygons, Histograms, Means,
Median & Mode
(b)
The histogram alongside shows the distribution of
times taken by a group of students to travel to school.
11 students took at least 5 but less than 10 minutes.
Complete the table.
Time
(t minutes)
Number
of students
0≤t<5
5 ≤ t < 10
11
10 ≤ t < 30
Ans.
Q8.
1
1
and not
as high as the other bar.
2
6
(a)
The bar for 2001 is
(b)
0 ≤ t < 5, 5 ; 10 ≤ t < 30, 8
The numbers of goals scored in 20 football matches were.
(a)
(i)
5
0
5
4
1
0
5
5
1
3
4
5
0
0
5
5
3
2
5
4
Complete the table in the answer space.
Number of goals
Frequency
0
1
2
3
4
5
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TOPIC#01
Statistics – Bar Charts, Frequency Polygons, Histograms, Means,
Median & Mode
(ii)
Ans.
(b)
State the median.
(c)
Calculate the mean number of goals.
(a)
(i)
(ii)
(b)
Q9.
Using the axes in the answer space, represent the information as a bar chart.
Number of
goals
0
1
2
3
4
5
Frequency
IIII
II
I
II
III
IIII
4
2
1
2
3
8
III
Diagram
4
(c)
3.1
A farmer collected eggs from his hens each day.
The numbers of eggs he collected on five days were
4 , 1 , 7 , 6 ,
4
(a)
Find the median number of eggs.
(b)
Calculate the mean number of eggs.
(c)
The farmer collected x eggs on the sixth day. The mean number of eggs collected
over the six days was 4.5. Find the value of x.
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TOPIC#01
Statistics – Bar Charts, Frequency Polygons, Histograms, Means,
Median & Mode
4
(b)
2
5
4
x=5
Ans.
(a)
(c)
Q10.
Some children were asked how many television programmes they had watched on the
previous day. The table shows the results.
Number of programmes
watched
0
1
2
3
Number of children
7
3
1
y
(a)
If the median is 2, find the value of y.
(b)
If the median is 1, find the greatest possible value of y.
Ans.
(a)
y = 10
Q11.
Answer the whole of this question on a sheet of graph paper.
(b)
y=8
The ages of a sample of 40 students were recorded.
The results are given in the table below.
Age (x years)
8 < x ≤ 10
10 < x ≤ 11
11 < x ≤ 12
12 < x ≤ 14
14 < x ≤ 16
16 < x ≤ 19
7
8
6
10
3
6
Frequency
(a)
Using a scale of 1 cm to represent 1 year, draw a horizontal axis for ages from 8 to
19 years.
Using a scale of 1 cm to represent 1 unit, draw a vertical axis for frequency densities
from 0 to 8 units.
On your axes, draw a histogram to illustrate the distribution of ages.
(b)
In which interval does the median lie?
(c)
Calculate an estimate of the mean age of the students.
(d)
Calculate an estimate of the number of students who were under 13 years old.
(e)
One student is chosen at random from this sample of 40 students.
Write down the probability that this student is
(i)
under 8,
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TOPIC#01
Statistics – Bar Charts, Frequency Polygons, Histograms, Means,
Median & Mode
(ii)
(f)
over 16.
A second student is now chosen at random from the remaining 39 students.
Calculate the probability that one student is over 16 and the other is not over 16,
Give your answer as a fraction in its lowest terms.
Ans.
(a)
Diagram
(b)
11 < x ≤ 12
(d)
Students under 13 years = 26
(e)
(i)
(f)
P (one over and other not over 16) =
(c)
mean = 12.4
P (under 8) = 0
(ii)
P (over 16) =
6 3

40 20
17
65
Q12.
The lengths of 40 nails were measured.
Their lengths in centimeters are summarized in the table below.
(a)
Length (l cm)
Frequency
0<l≤4
14
4 <l≤8
18
8 < l ≤ 16
8
On the axes in the answer space, draw the histogram which represents this
information.
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TOPIC#01
Statistics – Bar Charts, Frequency Polygons, Histograms, Means,
Median & Mode
(b)
Calculate an estimate of the mean length of the nails.
Ans.
(a)
(b)
Q13.
One hundred children were asked how far they could swim.
mean length = 5.8 cm
The results are summarized in the table.
Distance (d meters)
0 < d ≤ 100
100 < d ≤ 200
200 < d ≤ 400
Number of children
30
50
20
(a)
The histogram represents part of this
information. Complete the histogram.
(b)
A pie chart is drawn to represent the
three groups of children.
Calculate the angle of the sector that
represents the group of 20 children.
Ans.
(a)
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(b)
72°
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TOPIC#01
Statistics – Bar Charts, Frequency Polygons, Histograms, Means,
Median & Mode
Q14.
(a)
Sweet packets contain sweets of different colours.
The number of yellow sweets in each of 25 packets was recorded.
The table below shows the results.
Number of yellow sweets
0
1
2
3
4
5
Frequency
8
5
5
4
2
1
(iii)
1.6
For this distribution.
(i)
write down the mode,
(ii)
write down the median,
(iii)
calculate the mean.
(i)
The mode is 0.
Ans.
(a)
Q15.
The table shows the number of cars owned by each of 25 families.
(a)
(ii)
The median is 1.
Draw a bar chart to represent the.
Information in the table.
(b)
(c)
Find
(i)
the median number of cars,
(ii)
the modal number of cars,
(iii)
the mean number of cars.
0
3
4
1
0
1
1
2
3
2
3
6
1
0
1
2
0
3
2
3
4
1
2
1
A family is chosen at random.
Find the probability that it owns 3 cars.
(d)
2
Two families are chosen at random.
Find the probability that one family owns
2 cars and the other owns 4 cars.
(e)
A car is chosen at random.
Find the probability that it belongs to a family which owns 2 cars.
Ans.
(a)
Diagram
(b)
(i)
Median number of cars =2
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TOPIC#01
Statistics – Bar Charts, Frequency Polygons, Histograms, Means,
Median & Mode
(c)
Q16.
(ii)
Median number of cars = 1
(iii)
Mean number of cars = 1.92
1
5
(d)
1
25
(e)
1
4
The diagram shows a gauge for measuring the water level in a reservoir.
Readings, in meters, taken over a: certain period were as follows:
-2.3, -1.6, -0.4, 0.1, -0.5, 0.3, -1.2.
For the readings
(a)
find the difference, in meters, between
The highest and lowest levels,
2.6 m
(b)
find the median,
(c)
calculate the mean.
Ans.
(a)
(b)
-0.5 m
Q17.
On a certain stretch of road, the speeds
Of some cars were recorded.
The results are summarized in the table.
Part of the corresponding histogram is shown below.
(a)
(b)
Find the value of
(i)
p,
(ii)
q.
(c)
-0.8 m
Speed (x
km/h)
Frequency
25 < x ≤ 45
q
45 < x ≤ 55
30
55 < x ≤ 65
p
65 < x ≤ 95
12
Complete the histogram.
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TOPIC#01
Statistics – Bar Charts, Frequency Polygons, Histograms, Means,
Median & Mode
Ans.
(a)
(i)
p = 4 x 10 = 40
(ii)
q = 0.9 x 20 = 18
(b)
Q18.
(a)
In a survey, some students were asked which of
Three pictures, labeled X, Y and Z, they preferred.
The results are represented in the pi chart.
(i)
Calculate the percentage of students who
preferred X.
(ii)
Find, in its simplest form, the ratio of the
number of students who preferred X to those
who preferred Y.
(iii)
Given that 44 students preferred Y, calculate the number of students who
took part in a survey.
Ans.
Q19.
(a)
(i)
∴ 60% of students preferred X
(ii)
9 : 4
(iii)
165 students
Emma noted the number of letters in each of the 25 words in an examination question.
Number of letters
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2
3
4
5
6
7
8
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TOPIC#01
Statistics – Bar Charts, Frequency Polygons, Histograms, Means,
Median & Mode
The results are given in the table below.
Frequency
2
(a)
6
5
5
4
0
3
For this distribution,
(i)
write down the mode,
(ii)
find the median,
(iii)
calculate the mean.
(i)
mode = 3
Ans.
(a)
(ii)
median = 4 (iii)
Q20.
In an experiment, the heights of some plants were measured.
mean = 4.6
The table below summarizes the results.
Height (h cm)
Frequency
2<h≤3
3<h≤4
4<h≤5
5<h≤8
15
25
20
15
Complete the histogram which represents this information.
Ans.
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