Statistics Quiz (April 2015) Level IIName:Write in dark blue or
black pen. You may use a soft pencil for any diagrams or graphs.
Answer ALL questions and show the working below that questions when
required. The number of marks is given in brackets [ ] at the end
of each question or part question.1. On one day an insurance
company received 42 claims for storm damage. There were only
sufficient staff available to investigate six of these claims. In
order to select the six claims to be investigated, they were
randomly numbered 00 to 41, and then various methods of sampling
were suggested for selecting the six, using a table of two-digit
random numbers.
In selecting the random numbers, all repeats are ignored. The
suggested methods are
A Select six random numbers, ignoring any greater than 41.
B Select six random numbers. Divide each one by 42, and choose
the claims whose numbers correspond to the remainders, (for
example, if the selected random number is 45, claim 03 would be
chosen.)
C As in B, but ignore random numbers of 84 and over.
D Select a number at random from the range 00 to 06. Choose the
claim corresponding to that number, and every seventh number
thereafter, (for example, if 05 is selected, choose the claims
numbered 05 12 19 26 33 and 40).
E Do not use random numbers, just select the six largest
claims.
(i) Give the name of the method of sampling described in A, and
the name of that described in D.
Method A
...............................................................................
Method D
..........................................................................
[2]
(ii) State which of the five suggested methods of sampling are
unbiased, and which are biased.
Unbiased
...............................................................................
Biased
...........................................................................[2]
(iii) For each of the methods which is biased, give a reason why
it is biased.
...................................................................................................................................................
...................................................................................................................................................
...................................................................................................................................................
...............................................................................................................................................[2]
2
2. Twelve O-level Statistics students attended a revision course
shortly before taking their examination. They all took a test
before the start of the course, and another at the end of the
course. Their improvement during the course was measured as the
difference between their two test scores, (end mark minus start
mark). A negative value means that the end mark was lower than the
start mark. The differences were as follows.
13156239073811114
For each of the following state whether it is true or false.
(i) The median difference is 3.5.
............................[1]
(ii) The modal difference is 23.
............................[1]
(iii) The standard deviation of these differences cannot be
calculated because some of the values are negative.
............................[1]
(iv) The range is not an appropriate measure of dispersion for
the data because of the presence of one extreme value.
............................[1]
(v) The mean of these values is calculated by summing them and
then dividing the total by 12.
............................[1]
(vi) The range of the values is 1.
............................[1]
3
3 In a city there are three hospitals, and any person requiring
treatment at an Accident and Emergency department is equally likely
to attend any of the three.
The number of people attending each hospital, in thousands, to
the nearest thousand, in each of the years 2007 and 2008 is given
in the following table.
Hospital
Year
ABC
2007101416
2008121810
(i) Draw, on the grid below, a dual bar chart for each hospital
to illustrate the data.
[2]
4
(ii) Draw, on the grid below, a sectional bar chart for each
year to illustrate the data.
[2]
(iii) For each chart state one feature of the data which it
illustrates.
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
5
5 The table below gives the population (in millions, correct to
1 decimal place) of each of the four countries of the United
Kingdom at the Census in the year 2001.
CountryPopulation (millions)
England49.1
Scotland5.1
Wales2.9
Northern Ireland1.7
TOTAL58.8
The data are to be illustrated by a pie chart.
(i) Calculate, each to the nearest degree, the sector angles of
the pie chart.[2]
(ii) Using a circle of radius 5 cm, draw the pie chart.[2]
In the Census in the year 1951 the population of the United
Kingdom was 50.3 million (correct to 1 decimal place).
(i) Calculate, to 2 significant figures, the radius of the
comparable pie chart which could be used to represent the
population in 1951. [2]
6
6 The table below summarises how many O level subjects at grade
C were obtained by each of the 120 pupils who sat the examinations
at one school in a particular year.
Number of subjects0123456789
Number of pupils2211172425221241
For example, 17 pupils each obtained 3 subjects at grade C.
(i)Tabulate the cumulative frequencies for these data.[2]
(ii) Using a scale of 2 cm to represent 1 subject on the
horizontal axis, and 2 cm to represent a cumulative frequency of 10
on the vertical axis, draw an appropriate cumulative frequency
graph to illustrate these data.[4]
7
6 The cumulative frequency polygon below shows the weekly wage
of 60 women working for a company.
60
50
40
Cumulative frequency
30
20
10
0
0100200300400500600
Weekly wage ($)
Use the diagram to estimate
(i)the proportion of women earning a weekly wage of more than
$320,[2]
(ii)the median weekly wage,[1]
8
(iii)the range between the 10th and the 90th percentile of the
weekly wage.[3]
The range between the 10th and the 90th percentile of the weekly
wage for the men working for the same company was $240.
(iv)Using this information, describe briefly how the wages are
different for the men and women working for this company. [1]
In a class test the marks of 10 pupils were
5 4 10 3 3 4 7 4 6 5.
Find, for this distribution of marks,
(i)the mean,[1]
(ii)the mode,[1]
(iii)the median,[1]
(iv)the range.[1]
9
3. The maximum daily temperature, in C, was recorded throughout
the month of April. The results are shown in the table below.
Temperature (T C)Number of days
4 < T 106
10 < T 126
12 < T 148
14 < T 176
17 < T 214
(ii) Calculate, showing your working and giving your answers
correct to 1 decimal place, an estimate of
(a)the mean temperature,[2]
(b)the standard deviation.[4]
(iii) Using 2 cm to represent 2 C on the horizontal axis,
starting at 4 C, and a column of height 8 cm to represent the 12 `
T 14 temperature group, draw on graph paper a histogram to
illustrate the distribution of the daily temperature for
April.[5]
10(iii)Write down the modal class of the distribution.[1]
(iv)Estimate, correct to 1 decimal place, the percentage of days
having a temperature greater
than 15 C.[2]
For the month of June, the mean maximum daily temperature was
14.9 C and the standard deviation was 2.7 C.
(v)Make two comments on the differences between the figures for
June and the corresponding figures for April. [2]
11
8 In this question calculate all fertility rates per thousand,
and to the nearest whole number.
The fertility rate is defined as the number of births per 1000
females.
The table below gives information about the female population
and births in the town of Bluedorf for the year 2010, together with
the standard female population of the area in which Bluedorf is
situated.
Age group ofBirthsPopulation of femalesAge groupStandard
population
femalesin age groupfertility rateof females (%)
Under 20112320025
20 30459225015
31 40488305020
Over 4076400040
(i) Calculate the crude fertility rate for Bluedorf.
[4]
(ii) Calculate the fertility rate for each age group and insert
the values in the table above.
[2]
(iii) Calculate the standardised fertility rate for
Bluedorf.
[4]
12
The table below gives information about Redville, another town
in the same area as Bluedorf, also for the year 2010.
Age group ofFertility ratePopulation of females
females(per 1000 females)in age group
Under 20323000
20 302251560
31 401801700
Over 40204950
(iv) Calculate the standardised fertility rate for Redville in
the year 2010, using the same standard population as for
Bluedorf.
[2]
(v) Find how many more births there were in Bluedorf than in
Redville in the year 2010.
[2]
The local government of the area in which Bluedorf and Redville
are situated wishes to limit population growth, but only has
sufficient funds for a publicity campaign on birth control in one
of these two towns.
(vi) State, with a reason, in which of these two towns the
campaign should be conducted.
..........................................................................................................................................
..........................................................................................................................................
............................................................................................................................................
[2]
13
6 (a) The following table shows the death rates for Seeton in
1990, together with its population and the standard population.
AgePopulationDeath rateStandard
groupper 1000population
0 2519 000245%
26 5517 000530%
56 6914 000817%
70 and over10 000288%
Calculate
(i)the total number of deaths in 1990,[2]
(ii)the crude death rate, giving your answer to 1 decimal
place,[3]
(iii)the standardised death rate.[3]
The crude death rate of another town, Exton, in 1990 was 7.8 per
thousand.
The standardised death rate of Exton was 7.2 per thousand.
(i) State, with a reason, which of the two towns offers the
better chance of a longer life. [2]
14
8 The head teacher of a school analysed the schools expenditure
on the three stationery items, paper, books and other stationery.
The following table gives certain information relating to these
items for the three years 1990 (taken as base year), 2000 and
2002.
Price relatives1990 weights
199020002002
Paper1001201254
Books1001501607
Other stationery1001251255
(i) Calculate, to one decimal place, a weighted aggregate index
for these costs for 2002, taking
1990 as base year and using the weights 4, 7 and 5 which were
determined in 1990.[4]
(ii)Suggest a way in which these weights may have been
determined.[1]
(iii) The total amount spent on the three items in 1990 was $28
000. Use the index number you calculated in (i) to estimate the
total amount spent in 2002, giving your result to the nearest
$1000.[2]
15
(iv) State what you can deduce from the values of the price
relative for the other stationery in the
years 2000 and 2002.[1]
(v) Calculate new price relatives for each of the three items in
2002, taking 2000 as base year,
and giving your results to the nearest whole number.[3]
The table below shows the actual amounts spent in the year
2000.
ItemAmount ($)
Paper15 000
Books20 000
Other stationery30 000
(vi) Using these amounts as weights, and your results from (v),
show that the weighted aggregate
index for 2002, taking 2000 as base year, is 103.1.[3]
(vii) Use your answer to (vi) to estimate the total amount spent
on the three items in 2002, giving
your result to the nearest $1000.[1]
(viii) Suggest a reason for the difference in your two estimates
of expenditure in 2002, using your
results from (iii) and (vii).[1]