1.
The table shows information about the numbers of fatal road accidents in Great Britain.
Fatal road accidents in Great Britain
Year
Quarter
Number of fatal road accidents (x)
2004
2005
1
2
3
4
1
2
3
4
703
825
801
892
740
727
818
916
(Source: Department of Transport)
(a)
Calculate the mean number of fatal road accidents per quarter.
...................................
(1)
For the numbers of fatal road accidents in the table
(b)
x2 = 5 196 408
Calculate the standard deviation of the numbers of fatal road accidents.
Give your answer to 1 decimal place.
.....................................
(2)
In the first quarter of 2006 the number of fatal road accidents was 720
A motoring organisation is going to calculate the mean number of fatal road accidents per
quarter for 2004, 2005 and the first quarter of 2006.
(c)
Without carrying out any more calculations compare this mean with the mean found in
part (a).
Give a reason for your answer.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(Total 5 marks)
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2.
The parking times in hours (p) for 118 cars in a car park are summarised in the table.
(a)
Hours (p)
Frequency (f)
0<p1
16
1<p2
24
2<p4
30
4<p6
24
6 < p  12
24
p > 12
0
Draw a histogram for the data.
Parking times of cars in a car park
0
5
10
Parking time (hours)
15
(3)
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(b)
p
Frequency (f)
0<p1
16
1<p2
24
2<p4
30
4<p6
24
6 < p  12
24
Totals
118
Work out an estimate for the mean parking time of the cars.
You may use the space in the table.
........................ hours
(3)
(c)
Work out an estimate of the standard deviation of these parking times.
Give your answer to 1 decimal place.
You may use fx2 = 2872.
........................ hours
(3)
(Total 9 marks)
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3.
In an experiment a psychologist records the times, x seconds, for 50 people to complete a
puzzle.
The results are summarised in the table.
x
Number of people (f)
0 < x ≤ 40
10
40 < x ≤ 60
9
60 < x ≤ 80
12
80 < x ≤ 100
9
100 < x ≤ 180
10
Total
You may use the space provided in the table to answer this question.
(a)
(i)
Work out an estimate for the mean time.
..........................................
(ii)
Show that the standard deviation for the times is approximately 40 seconds.
(5)
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(b)
Draw a histogram to represent the data in the table.
(3)
(c)
(i)
Shade the region in your histogram that is within two standard deviations of the
mean.
(ii)
Find the proportion of people represented by this region.
..........................................
(4)
It is claimed that the time to complete the puzzle is normally distributed.
(d)
Comment on the validity of this claim.
......................................................................................................................................
......................................................................................................................................
......................................................................................................................................
(2)
(Total 14 marks)
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4.
In an experiment a biologist records the length, x cm, of 125 worms.
The results are summarised in the table.
x
Number of worms (f)
0<x4
36
4<x6
26
6<x8
20
8 < x  10
12
10 < x  15
21
15 < x  20
10
Total
125
You may use the space provided in the table to answer this question.
(a)
(i)
Show that an estimate for the mean length of the worms is 7.1 cm.
(ii)
Work out an estimate for the standard deviation of the lengths of the worms.
(You may use fx2 = 9089.75)
.................................................
(6)
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(b)
Draw a histogram to represent the data in the table.
(3)
(c)
(i)
Shade the region in your histogram that represents worms whose length is less than
or equal to the mean length of the worms.
(ii)
Find the number of worms whose lengths are less than or equal to 7.1 cm.
................................................
(3)
(d)
Are the lengths of these worms normally distributed? Explain your answer.
........................................................................................................................
(1)
(Total 13 marks)
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5.
Stephanie recorded the time she took to travel to work on each of 50 days.
The table shows information about these times.
(a)
Time (x minutes)
Frequency (f)
20 < x ≤ 30
4
30 < x ≤ 38
9
38 < x ≤ 42
12
42 < x ≤ 50
18
50 < x ≤ 60
7
Calculate an estimate of the mean time Stephanie took to travel to work.
....................................... minutes
(3)
(b)
Calculate an estimate of the standard deviation of these times.
You may use Σfx2 = 91367
....................................... minutes
(2)
(Total 5 marks)
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6.
The table gives information about the age, and the minimum stopping distance at 40 kilometres
per hour, for each of 10 cars.
Car
Age of car
(months)
Stopping distance
(metres)
A
9
28.4
B
15
29.3
C
24
37.6
D
30
36.2
E
38
36.5
F
46
35.4
G
53
36.3
H
60
44.1
I
64
44.8
J
76
47.2
(Source: www.bized.co.uk)
(a)
Work out Spearman’s rank correlation coefficient for these data.
You may use the blank columns in the table to help with your calculations.
............................................................
(4)
(b)
Describe and interpret your answer to part (a).
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(Total 6 marks)
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7.
At one location on a river, the distance from the bank and the corresponding depth of water
were measured.
The table shows the results.
Distance from
bank (cm)
0
50
150
200
250
300
350
400
450
500
Depth (cm)
0
10
26
44
59
52
74
104
85
96
Rank of Distance
1
2
3
4
5
6
7
8
9
10
Rank of Depth
d
d2
(Source: Research Project)
(a)
Complete the table.
(2)
(b)
Calculate Spearman’s rank correlation coefficient for these data.
.....................................
(2)
(c)
Interpret your answer to (b).
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(Total 6 marks)
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8.
The Gross Domestic Product per person, or GDP per capita, is a measure of a country’s wealth.
The greater the GDP per capita, the greater the country’s wealth.
The table shows the GDP per capita and the life expectancy at birth for each of nine countries.
Country
GDP per capita
($)
Life expectancy
at birth (years)
Luxembourg
58 198
78.7
UK
29 483
78.4
Monaco
26 844
79.6
Uruguay
14 423
76.1
Seychelles
7711
71.8
Grenada
4916
64.5
St. Helena
2413
77.8
Haiti
1484
52.9
Comoros
657
62.0
(Data source: www.nationmaster.com)
(a)
Work out Spearman’s rank correlation coefficient for these data.
.................................
(3)
(b)
Interpret your answer to part (a).
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(Total 5 marks)
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9.
The scatter diagram shows the ages, x million years, and the volumes, y cm3, of eight skulls of a
particular type of ape.
250
200
Volume of
skull ( y cm3)
150
100
50
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Age of skull (x million years)
(a)
Describe the correlation between the volume of a skull and the age of a skull for this type
of ape.
.....................................................................................................................................
(1)
The table gives the ages, x million years, and the volumes, y cm3, of the eight skulls shown in
the scatter diagram.
(b)
x
2.5
1.8
0.8
2.4
1.6
0.5
0.4
1.2
y
34
80
170
20
80
180
210
130
Calculate the coordinates of the mean point ( x , y ) for these data.
(..................... , .....................)
(2)
(c)
On the scatter diagram
(i)
plot the point ( x , y ) ,
(ii)
draw the line of best fit.
(2)
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A skull of this particular type of ape has an age of one million years.
(d)
Find an estimate for the volume of this skull.
.................................... cm3
(1)
The skull of another ape is to be classified. It has an age of 1 million years and a volume of 75
cm3.
(e)
Discuss whether this skull is likely to be from the same type of ape.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(f)
Give a reason why your line of best fit may not be used reliably to predict the volume of a
skull with an age of 3 million years.
.....................................................................................................................................
.....................................................................................................................................
(1)
The equation of the line of best fit has the form y  ax  b
(g)
Use your line of best fit to find the value of a and the value of b.
a = ..............................
b = ..............................
(3)
(h)
Give a practical interpretation of the meaning of a.
.....................................................................................................................................
.....................................................................................................................................
(2)
(Total 14 marks)
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10.
The table gives the mean and standard deviation of the marks in three examinations.
The marks in each of these examinations are normally distributed.
Examination
Mean
Standard deviation
French
60
15
German
56
10
Science
50
12.5
Mary got a mark of 72 in the French examination and a mark of 66 in the German examination.
(a)
Calculate Mary’s standardised score for these two examinations.
Standardised French score ...................................
Standardised German score .................................
(3)
(b)
Did Mary do better in the French examination or in the German examination? Give a
reason for your answer.
......................................................................................................................................
......................................................................................................................................
(1)
In the Science examination Mary had a standardised score of –0.56
(c)
Calculate Mary’s mark in the Science examination.
.........................................
(2)
(Total 6 marks)
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11.
In order to be considered for a place on a mechanics course at a local college, Wing and Mia
took tests in English and Mathematics.
Each test had a maximum mark of 100.
The table shows some information about the tests.
Wing
Mia
Overall
Mean
Standard deviation
English mark
48
55
50
10
Mathematics mark
59
51
55
8
(a)
Calculate Wing’s standardised scores in
(i)
English,
.....................
(ii)
Mathematics.
.....................
(3)
Mia’s standardised score in English is 0.5 and in Mathematics it is –0.5.
(b)
What is meant by a negative standardised score?
........................................................................................................................
........................................................................................................................
(1)
(c)
Who do you think did best overall? Give a reason for your answer.
........................................................................................................................
........................................................................................................................
(1)
(Total 5 marks)
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12.
Talil took a History exam.
The marks in this exam are normally distributed with a mean mark of 65 and a standard
deviation of 10
Talil had a mark of 52 in the History exam.
(a)
Calculate Talil’s standardised score for the History exam.
.......................................
(3)
Talil also took a Science exam.
His standardised score for the Science exam was – 0.25
(b)
Did Talil do better in the History exam or in the Science exam?
Give a reason for your answer.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(Total 5 marks)
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