Tag: construction of index numbers

Questions Related to construction of index numbers

Compute a price index for the following by simple aggregate method.

Commodity A B C D E F
Price in 1986 (Rs) 20 30 10 25 40 50
Price in 1991 (Rs) 25 30 15 35 45 55
business mathematics and statistics applied statistics weighted methods to calculate index numbers construction of index numbers index numbers

  1. 117.14

  2. 118.13

  3. 119.13

  4. 107.13

Show answer
Correct Option: A
Explanation:
 commodity  price in $1986$ $({p} _{0})$ price in $1991$ $({p} _{1})$ 
 A  $20$  $25$
 B  $30$  $30$
 C  $10$  $15$
 D  $25$  $35$
 E  $40$  $45$
 F  $50$  $50$


$\sum { {p} _{0} }$ = $175$ , $\sum {{p} _{1} }$= $200$
price index number ${p} _{01}$ = $\dfrac {\sum {{p} _{1} }}{\sum { {p} _{0} } } \times 100$
=$\dfrac{200}{175} \times 100$

=$ \dfrac{20000}{175}$

= $117.14$

Compute the consumer price index for 1990 taking 1989 as the base year.

Commodity Price in 1989 Price in 1990
Butter 20 21
Cheese 16 12
Milk 3 3
Eggs 2.80 2.80
business mathematics and statistics applied statistics weighted methods to calculate index numbers construction of index numbers index numbers

  1. 93

  2. 94

  3. 95

  4. 96

Show answer
Correct Option: A
Explanation:
 $Commodity$ $Price\,in\,1989$$P _0$  $Price\,in\,1990$$P _1$ 
 $Butter$ $20$  $21$ 
$Cheese$  $16$  $12$ 
$Milk$  $3$  $3$ 
$Eggs$  $2.80$  $2.80$ 
 $Total$ $\sum P _0=41.8$  $\sum P _1=38.8$ 

$\therefore$   By using simple aggregate method,

$\Rightarrow$  $P _{01}=\dfrac{\sum P _1}{\sum P _0}\times 100=\dfrac{38.8}{41.8}\times 100 =92.82 \approx 93$

Calculate cost of living index from the following table of prices and weights.

Commodity Weight Price index
Food 35 108.5
Rent 9 102.6
Clothes 10 97
Fuel 7 100.9
MIscellaneous 39 103.7
business mathematics and statistics index numbers weighted methods to calculate index numbers construction of index numbers applied statistics

  1. 104.4

  2. 106.5

  3. 126.5

  4. 128.5

Show answer
Correct Option: A
Explanation:
 $Commodity$ $Weight$$w$  $Price\,index$$I$  $I.w$ 
 $Food$ $35$  $108.5$  $3797.5$ 
$Rent$  $9$  $102.6$  $923.4$ 
$Clothes$  $10$  $97$  $970$ 
$Fuel$  $7$  $100.9$  $706.3$ 
$Miscellaneous$  $39$  $103.7$  $4044.3$ 
$Total$  $\sum w=100$    $\sum I.w=10441.5$ 

$\Rightarrow$   Cost of living index = $\dfrac{\sum I.w}{\sum w}=\dfrac{10441.5}{100}=104.4$

Calculate weighted index number for 2001 from the following data:

Item A B C
Quantity 20 15 10
Price in 2000 200 100 20
Price in 2001 320 120 28
business mathematics and statistics index numbers weighted methods to calculate index numbers construction of index numbers applied statistics

  1. 134.56

  2. 142.22

  3. 148.77

  4. 150.78

Show answer
Correct Option: B
Explanation:
 $Item$ $Quantity$$w$  $Price\,in\,2000$$P _0$  $Price\,in\,2001$$P _1$  $I=\dfrac{P _1}{P _0}\times100$ $Iw$ 
 $A$ $20$  $ 200$ $320$  $160$  $3200 $
$B$ $15$  $100$  $120$  $120$  $1800$ 
$C$  $10$  $20$  $28$  $140$  $1400$ 
 $Total$ $\sum w=45$        $\sum Iw=6400$

$\therefore$   By using weighted average price relative method.

$\Rightarrow$  $P _{01}=\dfrac{\sum Iw}{\sum w}=\dfrac{6400}{45}=142.22$ 

Taking 1975 as the base year with an index number 100 , calculate an index number for 1985 based on weighted average of price relatives.

Commodity A B C D
weight 20 30 10 40
Price per unit in 1975 10 20 5 40
Price per unit in 1985 30 35 10 80
business mathematics and statistics index numbers weighted methods to calculate index numbers construction of index numbers applied statistics

  1. 212.5

  2. 217.5

  3. 219.5

  4. 345.65

Show answer
Correct Option: A
Explanation:
$Commodity$ $Weight$$w$  $Price\,in\,1975$$P _0$  $Price\,in\,1985$ $P _1$ $Price\,relative $$I=\dfrac{P _1}{P _0}\times 100$ $I.w$ 
 $A$ $20$  $10$  $30$  $300$  $6000$ 
$B$  $30$  $20$  $35$  $175$  $5250$
$C$ $10$  $5$  $10$  $200$  $2000$ 
$D$ $40$  $40$  $80$  $200$  $8000$ 
 $Total$ $\sum w=100$        $\sum I.w=21250$

$\Rightarrow$  By using weighted average of price relative method,

$\Rightarrow$  $P _{01}=\dfrac{\sum I.w}{\sum w}=\dfrac{21250}{100}=212.5$

The quotations for four different commodities for the years 2000 and 2005 are given below. Calculate the index number for 2005 , with 2000 as base year by using weighted average of price relatives method.

business mathematics and statistics index numbers weighted methods to calculate index numbers construction of index numbers applied statistics

  1. $164.05$

  2. $161.05$

  3. $154.05$

  4. $151.05$

Show answer
Correct Option: A

Calculate the cost of living index(approximately) from the following data:

Group Weights Group Index No.
Food 47 247
Fuel and Lightning 7 293
Clothing 8 289
House Rent 13 100
Miscellaneous 14 236
business mathematics and statistics index numbers weighted methods to calculate index numbers construction of index numbers applied statistics

  1. $231.2$

  2. $265.4$

  3. $245.7$

  4. $123.78$

Show answer
Correct Option: A
Explanation:

Cost of living index $=\cfrac { \sum _{ i=1 }^{ 5 }{ { \left( \text{weight }\right)  } _{ i } } \times { \left( \text{Index no.} \right)  } _{ i } }{ \sum _{ i=1 }^{ 5 }{ { \left( \text{weight} \right)  } _{ i } }  } $
$=\cfrac { 47\times 247+7\times 293+8\times 289+13\times 100+14\times 236 }{ 47+7+8+13+14 } $
$=\cfrac { 11609+2051+2312+1300+3304 }{ 89 } $
$=\cfrac { 20756 }{ 89 } =231.19$
$\simeq 231.2$

Using simple aggregate method, calculate price index number from the following data:

Commodity A B C D
Price in 1997 90 40 90 30
Price in 1998 95 60 110 35
business mathematics and statistics applied statistics weighted methods to calculate index numbers construction of index numbers index numbers

  1. 110

  2. 120

  3. 130

  4. 140

Show answer
Correct Option: B
Explanation:
 $Commodity$  $Price\, in\, 1997$          $P _0$  $Price\, in\, 1998$       $P _1$
 $A$ $90$ $95$
 $B$ $40$  $60$ 
 $C$ $90$  $110$
 $D$ $30$ $35$ 
 $Total$  $\sum P _0=250$  $\sum P _1=300$

$\Rightarrow$  Price index number $(P _{01})$ = $\dfrac{\sum P _1}{\sum P _0}\times 100=\dfrac{300}{250}\times=120$

Construct a composite index number from the following index numbers and weights:

Index Numbers 127 142 186 172 115
Weight 5 4 3 6 8
business mathematics and statistics index numbers weighted methods to calculate index numbers construction of index numbers applied statistics

  1. $134$

  2. $145$

  3. $143$

  4. $149$

Show answer
Correct Option: C
Explanation:

Composite index no. $=\cfrac { \sum _{ i=1 }^{ 5 }{ { \left( \text{index no. }\right)  } _{ i } } \times { \left( \text{Index no.Weight} \right)  } _{ i } }{ \sum _{ i=1 }^{ 5 }{ { \left(\text{ weight }\right)  } _{ i } }  } $
$=\cfrac { 127\times 5+142\times 4+186\times 3+172\times 6+115\times 8 }{ 5+4+3+6+8 } $
$=\cfrac { 635+568+558+1038+920 }{ 26 } $
$=\cfrac { 3713 }{ 26 } =142.8$
$\simeq 143$

The following commodities have the given price indices relative to a base of $100$. The weights are also given:

Commodity Relative Index Weight
Butter 181 4
Bread 116 12
Tea 110 3
Bacon 152 7

Calculate the new index for this set of commodities

business mathematics and statistics index numbers weighted methods to calculate index numbers construction of index numbers applied statistics

  1. $132$

  2. $133$

  3. $134$

  4. $135$

Show answer
Correct Option: D
Explanation:

New index $=\cfrac { \sum _{ i=1 }^{ 4 }{ { \left( \text{Relative index }\right)  } _{ i } } \times { \left( \text{Index no. Weight }\right)  } _{ i } }{ \sum _{ i=1 }^{ 4 }{ { \left( \text{weight }\right)  } _{ i } }  } $
$=\cfrac { 181\times 4+116\times 12+110\times 3+152\times 7 }{ 4+12+3+7 } $
$=\cfrac { 724+1392+330+1064 }{ 26 } $
$=\cfrac { 3510 }{ 26 } =\cfrac { 1755 }{ 13 } $
$=135$