Tag: weighted methods to calculate index numbers

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

Laspeyre's index $= 110$, Paasche's index $= 108$, then Fisher's Ideal index is equal to: 

  1. $110$
  2. $108$
  3. $100$
  4. $109$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Laspeyre's Index $(L.I.)$ $= 110$

Paasche's Index $(P.I.)$ $= 108$

Fisher's Ideal Index $= \sqrt{L.I. \times P.I.}$
                                 $= \sqrt{110\times 108}$
                                 $=108.995 \approx 109$

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

If all the values are not of equal importance the index number is called: 

  1. Simple

  2. Unweighted

  3. Weighted

  4. None
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\Rightarrow$  If all the values are not of equal importance the index number is called : $Weighted$

$\Rightarrow$  The ratio of the sum of weighted prices of current and base time periods multiplied by 100 is called weighted aggregate price index.
$\Rightarrow$  This index is calculated after allocating weight to each commodity on the basis of their relative importance.
$\Rightarrow$  Weight of these commodities are then multiplied by the prices of base and current time periods. these prices are called weighted price.

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

The aggregative expenditure method and family budget method always give:

  1. Different results

  2. Approximate results

  3. Same results

  4. None of them
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\Rightarrow$  The aggregative expenditure method and family budget method always give : $Same\,\,result.$

$\Rightarrow$  Aggregate Expenditure Method - In this method, the quantities of commodities consumed by the particular group in the base year are estimated and these figures or their proportions are used as weights.
$P _{0n}=\dfrac{\sum P _n q _0}{\sum P _0 q _0}\times 100$
Here, $P _n$ Represent the price of the current year,
$P _0$  Represents the price of the base year and
$q _0$  Represents the quantities consumed in the base year.
$\Rightarrow$  Family Budget Method - In this method, the family budgets of a large number of people are carefully studied and the aggregate expenditure of the average family for various items is estimated. These values are used as weights.
$P _{0n}=\dfrac{\sum WI}{\sum W}$    Here, $I=\dfrac{P _n}{P _0}\times 100$  and $W=P _0 q _0$

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

A factory uses three raw materials A ,B and C in the manufacturing process.The price of material were as shown below: Calculate a simple aggregate index for $2005$.

Commodity Price in Rs in 1995 Price in Rs in 2005
A 4 5
B 60 57
C 36 42      

  1. 119

  2. 106

  3. 104

  4. 108
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

<

 Commodity Price in Rs in $1995$ $({p} _{0})$ Price in Rs in $2005$ $({p} _{1})$
Commodity
 A  $4$  $5$
 B  $60$  $57$
 C  $36$ $42$


$\sum { {p} _{0} }$ = $100$ , $\sum {{p} _{1} }$= $104$
price index number ${p} _{01}$ = $\dfrac {\sum {{p} _{1} }}{\sum { {p} _{0} } } \times 100$
=$\dfrac{104}{100} \times 100$
= $104$

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

The most appropriate average in averaging the price relatives is:

  1. Median

  2. Harmonic mean

  3. Arithmetic mean

  4. Geometric mean
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\Rightarrow$  The most appropriate average in averaging the price relative is : $Geometric\,\, mean.$

$\Rightarrow$  The geometric mean is the average of a set of products, the calculation of which is commonly used to determine the performance results of an investment or portfolio. It is technically defined as "the 'n'th root product of 'n' numbers.
$\Rightarrow$  The geometric mean must be used when working with percentages, which are derived from values.

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

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

Commodity A B C D E
1993 prices (in Rs) 50 40 10 5 2
1995 prices(in Rs) 80 60 20 10 6

  1. 164.69

  2. 154.75

  3. 162.69

  4. 152.75
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
 $Commodity$  $Price\, in\, 1993$  $( In\, Rs.)$ $P _0$  $Price\, in\, 1995$$(In\, Rs.)$ $P _1$
 $A$  $50$ $80$
 $B$ $40$  $60$ 
 $C$ $10$  $20$ 
$D$  $5$  $10$
 $E$ $2$  $6$ 
 $Total$  $\sum\,P _0=107$  $\sum\,P _1=176$

$\Rightarrow$   Here, $\sum\,P _0=107$ and $\sum\,P _1=176$

$\therefore$      Simple Aggregate Price Index $P _{01}=\dfrac{\sum P _1}{\sum P _0}\times 100=\dfrac{176}{107}\times 100=164.48$

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

Calculate the index number for the year 2006 with 1996 as the base year by the weighted average of price relatives method from the following data.

Commodity A B C D E
Weight 40 25 5 20 10
Price(Rs) per unit 1996 32 80 1 10.24 4
Price(Rs) per unit 2006 40 120 1 15.36 3

  1. 130

  2. 133.34

  3. 138.34

  4. 139.45
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
 $Commodity$ $Weight$ $w$ $Price\,in\,1996$$(Rs.)\,\,P _0$  $Price\,in\,2006$$(Rs.)\,\,P _1$  $Price\,relative$$I=\dfrac{P _1}{P _0}\times 100$  $I.w$ 
$A$  $40$  $32$  $40$  $125$  $5000$ 
$B$  $25$  $80$  $120$  $150$  $3750$ 
$C$  $5$  $1$  $1$  $100$  $500$ 
$D$  $20$  $10.24$  $15.36$  $150$  $3000$ 
$E$  $10$  $4$  $3$  $75$  $750$ 
 $Total$  $100$       $13000$

$\Rightarrow$  $P _{01}=\dfrac{\sum Iw}{\sum w}=\dfrac{13000}{100}=130$

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

Calculate price index for the following by using price relative method.

Material Cement Timber Steel Bricks
Price in 1969 (in Rs) 5 9.5 35 12
Price in 1970 (in Rs) 8 14.3 42 24

  1. 152.34

  2. 135.5

  3. 157.5

  4. 154.25
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
 $Material$  $Price\, in\, 1969$$(in\,Rs.)\,[P _0]$  $Price\,in\,1970$$(in\,Rs.)\,[P _1]$ $Price\, Relative$$\dfrac{P _1}{P _0}\times 100$ 
$Cement$  $5$ $8$   $160.00$
$Timber$ $9.5$  $14.3$   $150.52$
$Steel$  $35$  $42$   $120.00$
$Bricks$ $12$  $24$   $200.00$
$Total$      $630.52$

$\Rightarrow$  $P _{01}=\dfrac{\dfrac{P _1}{P _0}\times 100}{N}=\dfrac{630.52}{4}=157.5$

$\therefore$    Price index for 1970, taking 1969 for base year = $157.5$

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

  1. 120

  2. 122

  3. 130

  4. 132
Reveal answer Fill a bubble to check yourself
B Correct answer
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 = $\dfrac{\sum Iw}{\sum w}=\dfrac{3048}{25}=121.92\approx 122$.

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

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.

  1. 104

  2. 98

  3. 100

  4. 102
Reveal answer Fill a bubble to check yourself
D Correct answer
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 $102$.