IB MATH STUDIES SL 1: IB-Style Review Questions for Test
Descriptive Statistics and The Normal Distribution
1.
The weights of 90 students in a school were recorded. The information is displayed in the following table.
(a)
Weight (kg)
Number of students
40 ≤ w < 50
7
50 ≤ w < 60
28
60 ≤ w < 70
35
70 ≤ w < 80
20
Write down the mid interval value for the interval 50 ≤ w < 60.
(1)
(b)
Use your graphic display calculator to find an estimate for
(i)
the mean weight;
(ii)
the standard deviation.
(3)
(c)
Find the weight that is 3 standard deviations below the mean.
(2)
(Total 6 marks)
2.
A random sample of 167 people who own mobile phones was used to collect data on the amount of time
they spent per day using their phones. The results are displayed in the table below.
Time spent per
day (t minutes)
Number of people
(a)
0  t 15
15  t  30
30  t  45
45  t  60
60  t  75
75  t  90
21
32
35
41
27
11
State the modal group.
(1)
(b)
Use your graphic display calculator to calculate approximate values of the mean and standard
deviation of the time spent per day on these mobile phones.
(3)
(c)
On graph paper, draw a fully labelled histogram to represent the data.
(4)
(Total 8 marks)
1
3.
(a)
State which of the following sets of data are discrete.
(i)
Speeds of cars travelling along a road.
(ii)
Numbers of members in families.
(iii)
Maximum daily temperatures.
(iv)
Heights of people in a class measured to the nearest cm.
(v)
Daily intake of protein by members of a sporting team.
The boxplot below shows the statistics for a set of data.
2
4
6
8
10 12 14 16 18 20
data values
(b)
(c)
For this data set write down the value of
(i)
the median;
(ii)
the upper quartile;
(iii)
the minimum value present.
Write down three different integers whose mean is 10.
(Total 6 marks)
4.
The age in months at which a child first starts to walk is observed for a random group of children from a
town in Brazil. The results are
14.3, 11.6, 12.2, 14.0, 20.4, 13.4, 12.9, 11.7, 13.1.
(a)
(b)
(i)
Find the mean of the ages of these children.
(ii)
Find the standard deviation of the ages of these children.
Find the median age.
(Total 6 marks)
2
5.
The birth weights, in kilograms, of 27 babies are given in the diagram below.
1
2
3
4
(a)
7,
1,
0,
1,
8,
2,
1,
1,
9
2,
3,
2,
key 1|7 = 1.7 kg
3,
4,
3,
5,
5,
7
5,
5,
7,
6,
8,
6,
9
7,
9
Calculate the mean birth weight.
(2)
(b)
Write down:
(i)
the median weight;
(1)
(ii)
the upper quartile.
(1)
The lower quartile is 2.3 kg.
(c)
On the scale below draw a box and whisker diagram to represent the
birth weights.
(2)
1
2
3
Weight (kg)
4
5
(Total 6 marks)
6.
A group of students has measured the heights of 90 trees. The class calculate the mean height to be x =
12.4 m with standard deviation s = 5.35 m. One student notices that two of the measurements, 44.5 m and
43.2 m, are much too big and must be wrong.
(a)
How many standard deviations away from the mean of 12.4 is the value 44.5?
The incorrect measurements of 44.5 m and 43.2 m must be removed from the data.
(b)
Calculate the new value of x after removing the two unwanted values.
(Total 8 marks)
3
7.
The numbers of games played in each set of a tennis tournament were
9, 7, 8, 11, 9, 6, 10, 8, 12, 6, 8, 13, 7, 9, 10, 9, 10, 11,
12, 8, 7, 13, 10, 7, 7.
The raw data has been organized in the frequency table below.
games
frequency
6
2
7
5
8
n
9
4
10
4
11
2
12
2
13
2
(a)
Write down the value of n.
(b)
Calculate the mean number of games played per set.
(c)
What percentage of the sets had more than 10 games?
(d)
What is the modal number of games?
(Total 8 marks)
8.
The cumulative frequency table below shows the ages of 200 students at a college.
Age
Number of Students
Cumulative Frequency
17
3
3
18
72
75
19
62
137
20
31
m
21
12
180
22
9
189
23–25
5
194
> 25
6
n
(a)
What are the values of m and n?
(b)
How many students are younger than 20?
(c)
Find the value in years of the lower quartile.
(Total 8 marks)
4
9.
An atlas gives the following information about the approximate population of some cities in the year
2000. The population of Nairobi has accidentally been left out.
City
Population in Millions
Melbourne
3.2
Bangkok
7.2
Nairobi
Paris
9.6
São Paulo
17.7
Tokyo
28.0
Seattle
2.1
The atlas tells us that the mean population for this group of cities is 10.01 million.
(a)
Calculate the population of Nairobi.
(b)
Which city has the median population value?
(Total 8 marks)
10.
A marine biologist records as a frequency distribution the lengths (L), measured to the nearest centimetre,
of 100 mackerel. The results are given in the table below.
Length of mackerel
(L cm)
Number of
mackerel
27 < L ≤ 29
2
29 < L ≤ 31
4
31 < L ≤ 33
8
33 < L ≤ 35
21
35 < L ≤ 37
30
37 < L ≤ 39
18
39 < L ≤ 41
12
41 < L ≤ 43
5
100
(a)
Construct a cumulative frequency table for the data in the table.
(2)
(b)
Draw a cumulative frequency curve.
interval.
Hint: Plot your cumulative frequencies at the top of each
(3)
(c)
Use the cumulative frequency curve to find an estimate, to the nearest cm for
(i)
the median length of mackerel;
(2)
(ii)
the interquartile range of mackerel length.
(2)
(Total 9 marks)
5
The heights of 200 students are recorded in the following table.
(a)
Height (h) in cm
Frequency
140 ≤ h < 150
2
150 ≤ h < 160
28
160 ≤ h < 170
63
170 ≤ h < 180
74
180 ≤ h < 190
20
190 ≤ h < 200
11
200 ≤ h < 210
2
Write down the modal group.
(1)
(b)
Calculate an estimate of the mean and standard deviation of the heights.
(4)
The cumulative frequency curve for this data is drawn below.
200
180
160
number of students
11.
140
120
100
80
60
40
20
0
(c)
140
150
160
170
180
height in cm
190
200
210
Write down the median height.
(1)
(d)
The upper quartile is 177.3 cm. Calculate the interquartile range.
(2)
(e)
Find the percentage of students with heights less than 165 cm.
(2)
(Total 10 marks)
6
12.
The table below shows the number and weight (w) of fish delivered to a local fish market one morning.
weight (kg)
frequency
cumulative frequency
(a)
(i)
0.50 ≤ w < 0.70
16
16
0.70 ≤ w < 0.90
37
53
0.90 ≤ w < 1.10
44
c
1.10 ≤ w < 1.30
23
120
1.30 ≤ w < 1.50
10
130
Write down the value of c.
(1)
(ii)
On graph paper, draw the cumulative frequency curve for this data. Use a scale of 1 cm to
represent 0.1 kg on the horizontal axis and 1 cm to represent 10 units on the vertical axis.
Label the axes clearly.
(4)
(iii)
Use the graph to show that the median weight of the fish is 0.95 kg.
(1)
(b)
(i)
The zoo buys all fish whose weights are above the 90th percentile.
How many fish does the zoo buy?
(2)
(ii)
A pet food company buys all the fish in the lowest quartile. What is the maximum weight of
a fish bought by the company?
(3)
(c)
A restaurant buys all fish whose weights are within 10% of the median weight.
(i)
Calculate the minimum and maximum weights for the fish bought by the restaurant.
(2)
(ii)
Use your graph to determine how many fish will be bought by the restaurant.
(3)
(Total 16 marks)
13.
After their manufacture, the engines produced for a make of lawn mower are filled with oil by a machine
that delivers an average of 270 mL of oil with a standard deviation of 0.7 mL. The amounts of oil
delivered are normally distributed.
(a)
How much oil will be delivered to a machine that receives two standard deviations below the
average?
(2)
(b)
Draw a diagram of this normal distribution indicating the mean and the increments of standard
deviations above and below the mean. Shade the region that represents an amount of oil between
268.9 mL and 271.7 mL.
(3)
(c)
Find the percentage of the engines that receive more than 270.7 mL of oil, without using a
calculator.
(2)
(d)
The engines that receive the bottom 6% of oil will not start. How much oil is required for an
engine to start?
(2) (Total 9 marks)
7