Exercises on Statistics I. Index numbers
Decide whether the following statements are true or false.
… The price index reflects only the average price change.
… Index numbers are distribution ratios.
… Quantity index numbers are in natural measurement.
… The equation value index = price index · quantity index always holds.
… The quantity index reflects the average change of quantity, and shows its impact on the change of value.
… If quantity does not change, each quantity index equals 1.
… Fisher indices must always be calculated.
… The aggregative value index happens to exceed the simple value indices (value relatives)
… The Laspeyres index is easier to calculate than the Paasche index.
… You cannot directly compare years with the Paasche index.
… It is not possible to have an index below 100.
… It is not possible to have an index below 0.
Test questions (There is only one right answer to each question).
1.
Both Laspeyres- and Paasche- index numbers are the weighted average of the ……..
A. simple index numbers
B. aggregate value data
C. value index numbers
2.
The Laspeyres- index numbers can be calculated as a(n) ………………… mean (if the weights are of the same period).
A. arithmetic
B. harmonic
C. geometric
3.
The Paasche- index numbers can be calculated as a(n) ……………… mean (if the weights are of the same period).
A. arithmetic
B. harmonic
C. quadratic
4.
The values to be averaged for the aggregate value index are the …………
A. value data.
B. simple value index numbers.
C. aggregate value index numbers.
5.
If the prices of different milk products in a shop changed by +21; +1.8; +1.4; +1.6; -
0.9%, then the aggregate price index can be ………..
A. 123%.
B. 91%.
C. 107%.
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Exercises on Statistics I. Index numbers
6.
In the weighted arithmetic mean form of the Laspeyres-price index, the weights are
………….
A. the value data of the base period.
B. the value data of the current period.
C. the quantity data.
7.
In the aggregate form of the aggregate quantity index the weights are ……….
A. the value data of the base period.
B. the value data of the current period.
C. the price data.
8.
In the weighted harmonic mean form of the aggregate value index the weights are
……………
A. the value data of the base period.
B. the value data of the current period.
C. the simple value indices.
9.
If the Laspeyres- and the Paasche- price index numbers equal 96.4%, and 99.1%, respectively, then the calculation of the Fisher- price index is …………….., and its value is …………………
A. unnecessary
B. necessary, 100.5%
C. necessary, 97.8%
10.
The current weighted price index is larger than the base weighted price index if
………….
A. the order of the commodities by the quantity relatives equals the order of the commodities by the price relatives.
B. the order of commodities by quantity relatives is the inverse of the order of the commodities by the price relatives.
C. the relation between the quantities and the prices happens accidentally.
11.
The weights used in Paasche’s formula belong to ................
A. the base period
B. the given period
C. to any arbitrary chosen period
D. none of the above
12.
Laspeyres index formula uses the weights of the .....................
A. base period.
B. current year.
C. average of the weights of a number of years.
D. none of the above.
13.
The RPI in 2007 was 115.2 and by 2008 it was 138.5 using 2006 as the base year.
This represents a rise in the price of goods by ......................
A. 15%
B. 20.2%
C. 23.3%
D. £11.8
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Exercises on Statistics I. Index numbers
14.
The turnover of a company in 2007 was £54.5m and in 2008 it was £75 m If the RPI has increased from 101.9 in 2007 to 115.2 by 2008 using 2006 as the base year, than the real change in turnover has been:
A. £20.5m
B. £16m
C. £11.8
15.
If the price index of bananas was 100 in 2007 and that for apples was 110, then the price of a kg of bananas was .....................
A. less than for apples.
B. more than for apples.
C. impossible to compare.
Exercise 1
In Cheaptown the monthly food expenses of a typical family were the following:
Product
Bread
Milk measurement kg liter quantity
18
32
2000 price
94
86
25
44
2004 quantity price
110
130
Salami dkg 300 74 380 81
Questions: a) Calculate how the following changed during these years:
1.
the expenses by product type separately and also in total!
2.
the volume of consumption by product type separately and also in total!
3.
the selling price by product type separately and also in total! b) Highlight the relations between the various indices! c) Calculate how much more the family spent on food in forint due to the price and/or the volume change!
Exercise 2
A car dealer offers three types of cars: Audi, Skoda , and Seat . In 2004 Audis cost 6 200 000
HUF, Skodas cost 2 500 000 HUF, and Seats cost 3 900 000 HUF. In 2004, 43% of the turnover came from the sales of Skodas and 23% derived from selling Seats. Due to the appreciation of the forint the prices of the cars dropped, thus in 2005 Audis were offered for
5 999 000 HUF, Skodas for 2 350 000 HUF and Seats for 3 590 000 HUF. Turnover grew by
12,5% i.e. by 417 178 000 HUF by 2005.
Questions: a) Construct a statistical table! b) Calculate the composite index numbers! c) Report on your findings in writing!
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Exercises on Statistics I. Index numbers
Exercise 3
Incomes of a news agent in 2004 and 2005 were the following:
2004
Quantity
(piece)
Income
HUF
Quantity
(piece)
2005
Income
HUF
Daily paper
Weekly paper periodicals 441 254 760 860 329 840
Question:
Analyze the change in the total income and in the sold quantity and price by using index numbers.
Exercise 4
A company produces three different products.
Products
1 763
945
697 420
284 620
1 540
1 180
Production value at current price, million HUF
2004 2005
746 320
347 220
Prices in 2005 as a percentage of prices of
2004
A
B
C
20
70
10
50
43
30
210
90
200
Total 100 123
Questions: a) Calculate and interpret the simple and composite value-, price- and quantity- index numbers. b) Decide and report in writing whether the Paasche- or the Laspeyres price index will be larger. Give your reasons.
Exercise 5
The export-import turnover by commodity groups:
Commodity groups Current price
(million USD)
849
Import - 1999
Previous year=100%
Quantity index* previous year=100%
88.4 111.1 Food, tobacco, beverages
Raw materials
Energy sources
Processed commodities
Machineries and transport vehicles
629
1 705
10 773
14 052
82.6
100.9
104.3
117.5
115.7
104.1
114.6
128.3
Total
*: simple quantity indices by commodity groups
Source: Export-import Statistical Annual Report HCSO 2000
Question:
Give a detailed analysis how the import and the influential factors changed!
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Exercises on Statistics I. Index numbers
Exercise 6
The revenues of Blockbuilders Ltd calculated at current price grew by 2.64% from 2004 to
2005, while calculating at base prices the revenues rose by 1.97%.
Questions: a) Name the type of the given data! b) Report on the revenues of the company in writing (if necessary, calculate the missing index numbers)!
Exercise 7
The turnovers of a department store were the following:
Products
Sales revenues in 2004
(thousand HUF)
Volume Prices in 2004 in the percentage of 2003
Foods 2 244 102 110
Electric goods
Clothing
1 530
3 190
100
110
102
116
Others 942 108 105
Question:
Calculate and interpret all the possible simple and composite value-, price-, and quantityindices!
Exercise 8
The export-import turnover by commodity groups:
Commodity groups Current price
(million USD)
1 995
Export - 1999
Previous Quantity index* year=100 % previous year=100%
Food, tobacco, beverages
Raw materials
Energy-sources
619
407
82.3
91.6
93.6
111.1
119.6
100.4
Processed commodities
Machineries and transport vehicles
7 682
14 309
102.1
119.8
111.4
130.3
Total
*simple quantity indices by commodity groups
Source: Export-import Statistical Annual Report HCSO 2000
Question:
Give a detailed analysis how the export and the influential factors changed!
Exercise 9
The sales revenues from selling 3 products in 2003 were the following:
Product 1 2 3
Turnover (million HUF) 100 208 315
The changes in the prices in 2004 compared to 2003 were the following:
Product 1: +2%; Product 2: -3%; Product 3: +1%.
The total value of sales revenues increased by 3.8% in 2004 compared to 2003.
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Exercises on Statistics I. Index numbers
Question:
Calculate the appropriate aggregate quantity index.
Exercise 10
A wine trading company sells three types of red wine. From 2000 to 2001 the price/liter of
„ Egri Bikavér ” grew from 480 HUF to 600 HUF, the price of „ Soproni Kékfrankos ” increased from 350 HUF to 440, while that of „ Kisk ő rösi Kadarka ” rose from 230 HUF to 260 HUF.
The turnover of each type grew by 10%. In 2001 half of the turnover came from the sales of
„ Egri Bikavér” while one fifth of the turnover derived from selling „ Kisk ő rösi Kadarka ”.
Questions: a) Construct a statistical table! b) Calculate how much (%) the sold quantity of the red wines changed by product separately and in total!
Exercise 11
Incomes of a news agent (in HUF) in 2004 and their change compared to 2002 are the following:
Newspaper
Magyar Nemzet
Népszabadság
TVR-weekly
RTV-magazine
HVG
Value of the turnover in 2004 (HUF)
840 000
972 000
460 000
380 000
670 000
Change 2002=100%
Price
+10
+9
+11
+8
+7
Quantity
+2
+7
-2
+1
+4
Questions: a) Calculate how many percentages the turnover, the price and the volume of each newspaper grew separately and also in total! Name the columns first. b) How much more or less could the newspaper agent realize in forint due to the change of volume and/or price? c) Give detailed calculations about the relations of indices and the forint differences!
Exercise 12
At a florist the prices of the pot and fresh flowers, as well as other gardening accessories grew by 9%, 12% and 15%, respectively from 2004 to 2005. The income from selling pot flowers increased by 10 %, the income from fresh flowers grew by 15%, but the income from other accessories dropped by 5 %. In 2004 the total revenues of the shop equaled 530 000 HUF, in
2005 it was 578 230 HUF. In 2005 42% of the turnover came from selling fresh flowers while
20% came from selling accessories.
Questions : a) Compile a statistical table based on the data given above! b) Calculate the aggregate index numbers! c) How much more (in forint) could the florist realize due to the rise in the price and/or in the sold quantity? d) Report on your findings in writing!
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Exercises on Statistics I. Index numbers
Exercise 13
The following information has been released about the turnover of a furniture store: the turnover of the textile and accessories grew by 1.021%, of the bathroom accessories by
1.52%, of the kitchen and dining room department went up by 1.3%, of the sitting room department grew by 1.66%, and the turnover of the sleeping room and offices grew by 5.7% and 2.93% respectively. The sold quantities grew at each department despite fluctuation in the value of departmental turnovers.
Question:
Name the above indices. Based on the given information, decide whether the statements are true or false! Give your reasons! a) The turnover of the department store grew in total. b) The price level of the department store has not changed. c) The aggregate value index equals 105.8%.
Exercise 14
The quantity relatives of the main product groups of a food store are the following: 102.3%,
105%, 100%, 98%, 101%, 103%, and 99.2% .
Question:
Choose the one that contains the right indices and interpret them! a) I v
= 150.3%, I b) c) I
I d) I v e) I v v v p
0
0
= 150.9%, I p
= 111.2%, I p
1
=98%,
=98%,
= 99.764%, I
= 101.6%, I p
0 p
0
=114%,
=98%,
=98%,
I
I
I
I
I q q
1
1 q
1 q
1 q
1
=106%
=97%
=101.8%
=101.8%
=106%
Exercise 15
The net, after-tax sales revenues of the LIBRA book publisher grew by 2.8% at current price and by 1.9% at base price from 2004 to 2005.
Question: a) Name the given indices! b) Calculate how the prices of the books grew in total!
Exercise 16
Fill in the cells of the following table according to the headings so that the conditions are fulfilled!
Name i p
(%)
I
0 q
< I
1 q i q
(%)
I
0
> q
I
1 q
I
0 q
= I
1 q
X product
Y product
Exercise 17
In a restaurant the following information is known about the data of the cold and warm kitchen’s performance:
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Exercises on Statistics I. Index numbers
Kitchen
Cold kitchen i q
(%)
110.5 i p
(%)
103.5
Warm kitchen 114.7 105.8
Question:
Decide without calculation which weighted price index is larger! Give your reasons!
Exercise 18
The following information is known about the important products of a toy shop:
Product
CD, videocassette
Lego
Games i p
(%)
116.2
120.4
111.9 i q
(%)
107.6
108.9
101.2
Costume borrowing 109.3 93.4
Question:
Based on the above information decide whether the following statements are true or false! a) The sold quantity of the above products has not changed. b) The aggregate price index for the four products is between 109 and 120 %. c) The value of the two weighted price indices is the same. d) The price level of the above products in the toy shop has not changed. e) The value of the base weighted quantity index is larger than that of the current weighted one. f) Sales revenues dropped.
Exercise 19
Your work for a furniture producing and trading company Your boss, who knows that you graduated in a business school where you studied business statistics, asks your help to make a detailed analysis about the sales figures of the latest periods so as to achieve more effective production and trade in the future.
We know the following about the furniture trade in the last two years:
From 2006 to 2007 20% less room furniture was sold, but due to this figure the company turnover dropped by only 4%. The volume of sold kitchen furniture dropped by 8%, which caused a 3.4% drop in the turnover. At the same time 10% less dining room furniture was sold but their sales even raised the turnover by 3.5%. In 2006 the sales revenues from the three product groups equaled 300m HUF, 500m HUF and 200m HUF respectively.
Questions: a.) Compile a statistical table including the data given above! b.) After calculating the relevant indices give a detailed written report on the aggregate value-, quantity-, and price- changes!
c.) Calculate to what extent (in forint) the price and sold quantity changes increased the company’s turnover!
Exercise 20
You work for an electronic goods producing and trading company. Your boss asks your help to make a detailed analysis about the sales figures of the latest periods so as to achieve more effective production and trade in the future.
The following information has been revealed about DVD sales in the last two years:
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Exercises on Statistics I. Index numbers
From 2006 to 2007 the price of the DVD players grew by 8%, thus the turnover of the company grew by 18.8% out of this product group. DVD recorders were 10% more expensive in 2007, which caused a 15.5% turnover increase. The portable DVD players cost 25% more, thus the same quantity was sold in both years.
In 2006 the sales revenues from the three product groups equaled 500M Ft, 400M Ft, and
100M Ft respectively.
Questions: a.) Compile a statistical table including the data given above! b.) After calculating the relevant indices give a detailed written report on the aggregate value-, quantity-, and price- changes!
c.) Calculate to what extent (in forint) the price and sold quantity changes increased the company’s turnover!
Exercise 21
Data about the turnover of Product A, B and C are the following:
Distribution of Value % of turnover in the Quantity % of the turnover in
Product turnover in the base year (%) current year compared to the base year the current year compared to the base year
A
B
30
45
113.3
117.3
110
115
C
Question:
25
Calculate and interpret the Fischer price index!
Exercise 22
108 108
A lighting bulb producing company makes three types of bulbs. The following information is known about the turnovers of the last two years:
Type of bulb
40 W
60 W
100 W 30.0
Turnover (million HUF)
2004
…
…
2007
15.4
…
39.0 change (%)
…
…
…
Change in price
(2004 = 100%)
+6
…
+15
Change in volume
(2004 = 100%)
-2.15
-8.00
…
In total … … +18.0 …
It is also known that the turnover grew by 9 million forint from 2004 to 2007.
…
Questions: a.) Fill in the missing data in the table! b.) In writing report on the aggregate price and quantity change of the products with the calculation of the relevant index numbers! c.) How did the quantity change influence the sales revenues of the company (in forint)?
Exercise 23
You work for a new-born and baby- caring products making and trading company. Your boss, who knows that you graduated in a business school where you studied business statistics, asks your help to make a detailed analysis about the sales figures of the latest periods so as to achieve more effective production and trade in the future.
The company sells panties in three different packages. The following figures are known about the last two years’ trading:
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Exercises on Statistics I. Index numbers
Pieces in a package
20 pieces
50 pieces
100 pieces
Income
2006 (M HUF) 2007 (M HUF) 2006 = 100%
10
…
…
13
18
…
…
…
+ 8
Price change
(2006 = 100%)
+ 30
+ 25
…
Quantity index
(2006 = 100%)
…
– 4
– 10
In total 50 … … … …
Questions: a.) Fill in the missing data in the table! b.) In writing report on the aggregate price and quantity change of the products with the calculation of the relevant index numbers! c.) How did the quantity- and the price- changes influence the sales revenues of the company
(in forint)? d.) Calculate to what extent (in forint) the price and sold quantity changes increased the company’s turnover!
Exercise 24
You work for a new-born and baby- caring products making and trading company. Your boss, who knows that you graduated in a business school where you studied business statistics, asks your help to make a detailed analysis about the sales figures of the latest periods so as to achieve more effective production and trade in the future.
The company sells feeding bottles in three sizes. The following figures are known about the last two years’ trading:
Size of the feeding bottles
200 ml
300 ml
500 ml
2006-ban
(M Ft)
4
…
10
Turnover
2007-ben
(M Ft)
…
7.02
…
Change 2006
= 100%
…
+ 17
…
Quantity change
2006 = 100%
…
+ 30
+ 10
Price index
(2006 = 100%)
– 5
…
– 2
In total … 22.17 … … …
Questions: a.) Fill in the missing data in the table and give detailed calculations! b.) In writing report on the aggregate price and quantity change of the products with the calculation of the relevant index numbers! c.) Calculate to what extent (in forint) the price and the sold quantity changes increased the company’s turnover!
Exercise 25
The earnings of a worker and the RPI of two consecutive years are shown in the following table:
Year
2005
£/week
180
RPI (%)
115.2
2006 225 126.1
Questions: a.) How much better off is he in real terms if the effect of inflation is taken into account?
(The RPI is based in January 1987). b.) What could he buy for 100£ in 2006? c.) Give the graphical representation of the real and nominal wage changes in the given years!
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