Calculating the Index for 2015 Using 2013 As the Base Year

Solutions Manual

This document contains solutions to all the practice questions that appear at the end of each chapter.

Chapter 2 Solutions

Year / Cost
(£000s) / Index
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015 / 50.1
65.3
68.6
72
76.6
78.3
88.7
90.5
99.3
112.9 / 100.0
130.3
136.9
143.7
152.9
156.3
177.0
180.6
198.2
225.3

Year / Cost (£) / Index (Year 1 = 100)
1 / 650 / 100
2 / 580 / 89.2
3 / 410 / 63.1

Calculating the index for 2015 using 2013 as the base year

Resource / Index (2013 = 100)
Ink / 138
Card / 160
Labour / 107

So card has increased the most (up 60%)

Calculating both indices for 2014 based on 2013

2013 / 2014
p0 / q0 / pn / qn / p0q0 / pnq0 / p0qn / pnqn
Ink / 0.55 / 4000 / 0.58 / 5000 / 2200 / 2320 / 2750 / 2900
Card / 0.05 / 115000 / 0.05 / 124000 / 5750 / 5750 / 6200 / 6200
Labour / 6.5 / 25 / 6.9 / 34 / 162.5 / 172.5 / 221 / 234.6
8112.5 / 8242.5 / 9171 / 9334.6

Laspeyres’ index = 82428112.5×100=101.6 Paasch’s index = 9334.69171=101.8

For 2015 based on 2013

2013 / 2015
p0 / q0 / pn / qn / p0q0 / pnq0 / p0qn / pnqn
0.55 / 4000 / 0.76 / 3000 / 2200 / 3040 / 1650 / 2280
0.05 / 115000 / 0.08 / 110000 / 5750 / 9200 / 5500 / 8800
6.5 / 25 / 7 / 30 / 162.5 / 175 / 195 / 210
8112.5 / 12415 / 7345 / 11290

Laspeyres’ index = 124158112.5×100=153.0 Paasch’s index = 112907345=153.7

From 2013 to 2014 there has been very little change as prices have not changed much. However during 2014 to 2015 prices for ink and card have shown a big increase.

2005 / 2015
p0 / q0 / pn / qn / p0q0 / pnq0 / p0qn / pnqn
Bread / 28 / 6 / 78 / 6 / 168 / 468 / 168 / 468
Milk / 20 / 15 / 150 / 12 / 300 / 2250 / 240 / 1800
Tea / 96 / 1 / 75 / 2 / 96 / 75 / 192 / 150
564 / 2793 / 600 / 2418
Laspeyres' / 495.21 / Paasche's / 403.00

1998 / 2011
p0 / q0 / pn / qn / p0q0 / pnq0 / p0qn / pnqn
Company A / 160 / 200 / 520 / 500 / 32000 / 104000 / 80000 / 260000
Company B / 350 / 650 / 265 / 250 / 227500 / 172250 / 87500 / 66250
Company C / 105 / 600 / 140 / 400 / 63000 / 84000 / 42000 / 56000
Company D / 53 / 100 / 159 / 200 / 5300 / 15900 / 10600 / 31800
327800 / 376150 / 220100 / 414050
Laspeyres' / 114.75 / Paasche's / 188.12

7. Simple index for year 2 using year 1 as the base year

Supervisor: 14/12 x 100 = 116.7

Skilled: 10/9 x 100 = 111.1

Unskilled: 7/6 x 100 = 116.7

For year 3

Supervisor: 15/12 x 100 = 125.0

Skilled: 11/9 x 100 = 122.2

Unskilled: 8/6 x 100 = 133.3

So unskilled increased wages the most.

Both indices very similar and show a steady increase in total wages over the 3 years.

£105,000 is worth 105,000×186.7214.8 = £91,264

So price has dropped by £140,000 – 91,264 = £48,736

Year / Turnover / RPI / Real T/O
2005 / 15.3 / 192.0 / 15.30
2006 / 10.3 / 198.1 / 9.98
2007 / 12.1 / 206.8 / 11.23
2008 / 15.2 / 214.8 / 13.6
2009 / 24.4 / 213.7 / 21.9
2010 / 34.7 / 223.6 / 29.8

10.

2003 / 2011
p0 / q0 / pn / qn / p0q0 / pnq0 / p0qn / pnqn
7.50
10.00
8.00
18.00 / 120
41
25
21 / 9.00
12.50
10.00
22.40 / 158
52
30
25 / 900
410
200
378 / 1080
512.5
250
470.4 / 1185
520
240
450 / 1422
650
300
560
sum / 1888 / 2312.9 / 2395 / 2932

Laspeyres’ 122.51 Paasche’s 122.42

11.

2006 / 2011
p0 / q0 / pn / qn / p0q0 / pnq0 / p0qn / pnqn
3.63
2.11
10.03
4.01 / 3
4
1
7 / 4.49
3.26
12.05
5.21 / 2
6
1
5 / 10.89
8.44
10.03
28.07 / 13.47
13.04
12.05
36.47 / 7.26
12.66
10.03
20.05 / 8.98
19.56
12.05
26.05
57.43 / 75.03 / 50.00 / 66.64

Laspeyres’ 130.65 Paasche’s 133.28

12(a)

Month / Price / Index
Aug Sept Oct Nov Dec
Jan / 155
143
120
139
165
162 / 100.00
92.26
77.42
89.68
106.45
104.52

(b) (i) August to January change = 104:52 - 100 = 4:52%

(ii) September to December change = 165-143143=15.38%

13.

Year / Average wage / Average
RPI / Wages deflated to 2008 values
2008
2009
2010 / 255.1
271.3
290.7 / 214.8
213.7
223.6 / 255.1
272.7
279.3

Real wages grew more between 2008 and 2009, due to the decline in the RPI.

14.

Year / Average RPI / House price / Real house price
2008
2010 / 214.8
223.6 / £162 000
£157 500 / £162 000
£153 030

Real drop in price = £11 179 at 2010 prices, i.e. 6.9%

(a) (b) (i) (ii) (iii)

% change
Year / Average
RPI / RPI (2008 = 100) / since
2008 / since
2009
2008
2009
2010 / 214.8
213.7
223.6 / 100
99.5
104.1 / -0.5
4.1 / 4.9

Chapter 3 Solutions

Year / Population / Sample
Total / Female / Male / Total / Female / Male
1 / 320 / 192 / 128 / 20 / 12 / 8
2 / 250 / 150 / 100 / 16 / 10 / 6
3 / 230 / 138 / 92 / 14 / 8 / 6
Total / 800 / 480 / 320 / 50 / 30 / 20

Need to ask council tax payers. A survey could be sent out with council tax demands but that may produce a bias sample as only those people with strong views will respond. Maybe better to send questionnaire to random sample and follow up with letters. Could also allow residents to use online questionnaire.

The survey will probably be in the form of a simple questionnaire, possibly web based. Not necessary to have a random sample for this type of survey so anyone who applied for tickets could be asked to complete the questionnaire.

4 Various issues including asking age (use ranges), Vague questions (question 5), no option for ‘other’ in question 6. Unlikely that respondents would know what socio-economic group they are in.

5 1, 2 and possibly 5 would have a sampling frame.

6 (a) The sample would be Steve, Kim, Chris, Jane, Stuart, Jill. Average age is 40.3 and 3 read the Mirror, one the Sun, one The Times and one the Express.

(b) As we have 6 males and 3 females we need a sample of 4 males and 2 females. If we separate out the two sexes and use the same random numbers for each group then we will have Steve, Chris, Stuart, Kim, Julie and Jill. Average age is 39.2 and two read the Sun, two the Mirror and one each of The Times and Telegraph.

7. Answers depend on sample chosen.

Target population would be all council tax payers. A good sampling frame would be a database of council tax payers in a local council region. Alternatively the electoral register could be used although this will contain people not paying council tax (e.g. children over 18 living with parents or at university).

• Not all supporters belong to the supporters club so their views will be ignored.

• A simple random sample may not contain the right proportion of male/female supporters or the different ages.

• A better target population may be all people who have ever watched a game (if this data is kept) or even all people in the local town.

• A stratified sample would be best as it would take into account male/female and ages issue.

• A systematic sample may also be possible as every nth person leaving the ground could be sampled.

10.

Probably use multi-stage sampling methods. Randomly select a number of council regions then randomly select a number of schools within each region. A random sample of school leavers could then be selected from each chosen school.

11.

The simplest and cheapest survey would be a quota sample of shoppers in the town centre. However, this would ignore people who don’t shop in the town. An alternative would be an advert in a local newspaper, but this would bias towards those who read the paper and have a strong opinion. The most expensive would use the electoral register to contact a stratified sample of voters in the town.

12 (a) Both stratified and quota sampling aim to produce a sample that represents the target population in terms of the proportion contained within relevant sub-groups. However, stratified sampling requires the use of a sampling frame and is therefore a probabilistic sampling method.

Although it would be possible to obtain a list of all students it would probably be sufficient to employ a quota sampling method as this will be much cheaper to administer. The proportions of male/female and age ranges should be easy to obtain, which would make the quota sample reasonably accurate.

(b) No, since all students would be on campus.

· Misleading or ambiguous questions. Could be overcome by a pilot study.

· Asking the wrong question.

· Asking personal questions.

· Asking leading questions.

· Making the questionnaire too long.

· Difficulty in obtaining information about the student population.

· Method used to conduct survey. Face-to-face will probably be most effective but this may be too expensive.

· Poor response rate. This can be improved by face-to-face interviews.

· Bias sample if only certain members of the population are included in the sample (e.g. if only interview respondents in a particular part of the university).

· Conducting survey during vacations or exam periods!

13.

(a)

· Randomly select 40 employees from the database of all 200 employees

· Choose every 5th employee from the list

· For the shop-floor department there are 80 employees which is 40% of the total so we need a random sample of 40% of 40 = 16 employees from shop-floor. The other departments are calculated in a similar way

Dept / Number / Proportion of total / Sample size
Shop floor / 80 / .4 / 16
Service engineers / 15 / .075 / 3
Quality control / 20 / .1 / 4
Marketing / 25 / .125 / 5
Accounts / 15 / .075 / 3
Personnel / 10 / .05 / 2
Administration / 25 / .125 / 5
Catering / 10 / .05 / 2
Total / 200 / 1 / 40

Randomly select the required number from each department

· For quote sampling a survey could be conducted in the works canteen using the above numbers as the quota for each department; i.e. once the required number of employees has been chosen from one department no one else from that department would be selected

14 (a)

· Target population – all first year students and all staff.

· Sampling frame – database of student and staff records.

· Stratified sampling – sample contains same proportion of relevant categories as population.

· Multi-stage sampling – hierarchical structure. If the students were based on different campuses it might be necessary to pick one or two of these campuses randomly to reduce travelling time.

(b) Possible questions are:

· ‘Are you student or staff?’ (It will probably be necessary to be able to compare the responses between staff and students.)

· How strongly are you in favour of a 3rd semester?’ using a Likert scale. (This will be better than asking for a ‘yes’ or ‘no’ answer.)

· ‘How much would you be prepared to pay/receive to do a 3rd semester?’ (Will need to ascertain how financially viable the idea is. Students will need to pay in order for lecturing staff to be paid.)

· ‘At what faculty are you based?’ (There may be differences between faculties.)

· For students only – ‘What is your age (within a range)?’ (Older students might be more interested in completing their degrees quicker.)

(c) A postal or email questionnaire will probably be the best method here. Alternatively, could be face-to-face if quota sampling used.

15. There are many reasons such as:

· The bins are being refilled when empty so difficult to judge how empty the bin is.

· Were all the bins filled to the same level?

· Do people prefer different colours?

· Positioning of the bins might make a difference. Do you take the nearest regardless of your views?

· Do people associate colour with flavours?

· And, of course, this was not a random sample!

Chapter 4 Solutions

1.(a) Continuous as there could be a fraction of an hour. However if the answer is obtained by counting the number of hours of sunshine then it could be classed as discrete.

(b) Continuous

(d) Ordinal

2. (a) Time spent online

(b) Download speed

(d) Rating given to a site

(e) Choice of social networking site

Class interval / Frequency
0 to 2 / 5
3 to 5 / 4
6 to 8 / 2
9 to 11 / 2
12 to 14 / 3
15 to 19 / 0
20 to 30 / 2

A bar chart could be drawn as below

This bar chart is not very good as it doesn’t take into account the different class intervals. (See the solution to question 8 for a better chart).