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]

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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]

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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]

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(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.

...........................................................................................................................................

...........................................................................................................................................

...........................................................................................................................................

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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]

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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]

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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]

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(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]

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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]

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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]

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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]

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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]

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(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]