Tag: business mathematics and statistics

Questions Related to business mathematics and statistics

Construct a composite index number as a weighted mean from the following data:

Index Number 122 145 101 98 137 116
Weight 7 2 4 1 6 5
business mathematics and statistics index numbers weighted methods to calculate index numbers construction of index numbers applied statistics

  1. 120

  2. 122

  3. 130

  4. 132

Show answer
Correct Option: B
Explanation:
 $Index\,number$        $I$ $Weight$$w$  $I.w$ 
 $122$ $7$  $854$ 
$145$  $2$  $290$ 
$101$  $4$  $404$ 
$98$  $1$  $98$ 
$137$  $6$  $822$ 
$116$  $5$  $580$ 
$Total$  $\sum w=25$  $\sum Iw=3048$ 

$\Rightarrow$  Composite index number = $\dfrac{\sum Iw}{\sum w}=\dfrac{3048}{25}=121.92\approx 122$.

A firm uses three raw materials E ,F ,G in processing . The price per kg of these materials are as shown:

Item 1957 1967
E 4 3
F 60 57
G 36 42

Calculate simple aggregate price index for 1967 using 1957 as the base year.

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

  1. 104

  2. 98

  3. 100

  4. 102

Show answer
Correct Option: D
Explanation:
 $Item$  $Price\,in\,1957$         $P _0$  $Price\,in\,1967$         $P _1$
 $E$ $4$  $3$ 
$F$  $60$  $57$ 
$G$  $36$  $42$ 
$Total$ $\sum P _0=100$  $\sum P _1=102$

$\Rightarrow$ $\sum P _{01}=\dfrac{\sum P _1}{\sum P _0}\times=\dfrac{102}{100}\times 100=102$

$\Rightarrow$  The price index for year $1967$, taking $1957$ base year is $102$.

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