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Statistics 4 Marks Each

INDEX NUMBER · Gujarat Board (GSEB) STD 12 Commerce (GSEB - English Medium). 27 questions.

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27 questions · timed · auto-graded

Question 14 Marks
The data about the purchase of groundnut by an edible oil mill from the year $2008$ to $2015$ are as follows. Prepare the index numbers by fixed base method with the year $2008$ as the base year, with chain base and by taking the average quantity purchased in the year $2010$ and $2011$ as the purchase for the base year.
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Question 24 Marks
Find the real wages for the worker class of a city from the following data about their average monthly wage and the cost of living index number $($base year $2001).$ Find the purchasing power of money in the year $2015$ by taking the base year $2001$ and state the importance of this answer.
Answer
The calculation of real wage using the wages and cost of living index numbers will be as follows.
Real wage $=\frac{\text { Average monthly wage }}{\text { Cost of living index number }} \times 100$
l‘he purchasing power of money is the reciprocal of the cost of living index number of the current year with the respective base year.
$\therefore$ We can say that the purchasing power of money in the year $2015$ with the base year $2001=\frac{100}{287}=$ $0.3484 \simeq 0.35$.
Hence, it can be said that if the unit of money is rupee then the value of rupee in the year $2005$ is $35$ paise with respect to the base year $2001 .$
Thus, although the actual average monthly wage of the workers of this class in the year $2015$ is more than the base year $2001,$ the real disposable wage in the year $2015$ is only $₹ 696864$ with respect to the base year.
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Question 34 Marks
The following data are obtained from the budget inquiry of middle class families. State the change in the cost of living in the current year $2015$ with respect to the base year $2014$ by finding the index number. If the average monthly disposable income of a family is f $30,000$ during the year $2014$ and their average monthly disposable income during the year $2015$ is $?\ 35,000$ then according to family budget index number, what should be the rise in the average monthly disposable income of the family to maintain the same standard of living of the base year ?
Answer
The weights $W$ and the percentage price relatives $1$ of the groups for the year $2015$ are given here.
Hence, we shall calculate the index number by family budget method.
Index number by family budget method $I=\frac{\Sigma I W}{\Sigma W}$ $ =\frac{13450}{100}=134.5$
Thus, it can be said that there is a rise of $(134.5 — 100) = 34.5\% $ in the cost of living in the year $2015$ as compared to the year $2014.$
According to the family budget method index number of the year $2015,$
the average monthly disposable income to maintain the same standard of living as the base year
$=\frac{\text { Index number of current year }}{\text { Index number of base year }} \times$ income of base year$=\frac{134.50}{100} \times 30,000$$=₹ 40,350$
The required increase in the average monthly disposable income to maintain the standard of living ofthe family $= ₹ 40,350 - ₹ 35,000 = ₹ 5350$
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Question 44 Marks
The data referring to worker class of a city are as follows. Find the general index numbers for the years $2014$ and $2015.$ If the wages of these workers in $2014$ are increased by $5 \%$ in the year $2015,$ is this rise in wages sufficient to maintain their standard of living $?$
Answer

Index number for year $2014=\frac{\Sigma I_{1} W}{\Sigma W}=\frac{21060}{100}=210.60$
Index number for year $2015=\frac{\Sigma I_{2} W}{\Sigma W}=\frac{22540}{100}=225.40$
There is a rise of $(225.4- 210.6) = 14.8 \%$ in the cost of living of workers in the year $2015$ than in the year $2014$ with reference to the base year.
Thus, the percentage increase in the cost of living index number in the year $2015$ is $\frac{14.8}{210.6} \times 100=7.03$ as compared to the year $2014.$
Hence, the rise of $5 \%$ in the wages of the year $2014$ is not sufficient to maintain the same standard of living of the workers in the year $2015 .$
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Question 54 Marks
Calculate the cost of living index number by the total expenditure method and the family budget method for the year $2015$ with the base year $2014$ using the following data.
Answer
The base year is $2014.$ We will take $p_{0}=$ price of $2014, q_{0}=$ quantity of $2014$ and $p_{1}=$ price of $2015.$ We shall make uniform units for the price and quanity of each item.
Method of Total Expenditure

$ \text { Index number by total expenditure method } =\frac{\sum p_{1} q_{0}}{\sum p_{0} q_{0}} \times 100$
$ =\frac{6480}{4580} \times 100$
$ =141.4847$
$ \simeq 141.48 $
Thus, there is a rise $01"(141 .48 - 100) = 41.48 \%$ in the total expenditure in the year $2015$ as compared to the base year $2014.$
Method of Family Budget
$ \text { Index number by family budget method } =\frac{\Sigma I W}{\Sigma W}$
$ =\frac{648000}{4580}$
$ =141.4847$
$ \simeq 141.48 $
Note : We can see here that the index numbers obtained by total expenditure method and family budget method are same.
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Question 64 Marks
Find Fisher’s index number for the year $2015$ by taking the year $2014$ as the base year from the data given below about consumption and total expenditure of five different items.
Answer
The consumption and total expenditure for the items are given here.
Total expenditure of item $=($ Price of item per unit $) \times($ Consumption of item $)$
$\therefore$ Price of item per unit $=\frac{\text { Total expenditure of item }}{\text { Consumption of item }}$
We will obtain the price per unit of each item using the above formula.
${\text{Fisher's index number }} I_{F} =\sqrt{\frac{\sum p_{1} q_{0}}{\sum p_{0} q_{0}} \times \frac{\Sigma p_{1} q_{1}}{\sum p_{0} q_{1}}} \times 100$
$ =\sqrt{\frac{4850}{3830} \times \frac{5450}{4230}} \times 100$
$ =\sqrt{1.6315} \times 100$
$ =1.2773 \times 100$
$I_{F} \simeq 127.73 $
Thus, it can be said that there is $(127.73 - 100) = 27.73 \%$ rise in the prices in the year $2015$ as compared to the year $2014.$
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Question 74 Marks
Calculate the real wages of a worker class from the following data about their monthly wages. Find the purchasing power of money in the year $2015$ considering the year $2008$ as the base year.
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Answer

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Purchasing power of money:
Base year $2008,$ cost of living index number for the year $2015$ is $260$
$\therefore $ Purchasing power of money in the year $2015 = \frac{1}{\text { Cost of living index number }} \times 100$
$= \frac{1}{260} \times 100$
$= \frac{100}{260}=₹ 0.38 = ₹ 0.38$
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Question 84 Marks
The price of wheat increased by $70 \%$ and price of rice increased by $40 \%$ in the year $2015$ with respect to the year $2010.$ The price of bajri decreased by $25 \%.$ The price of oil increased by $40 \%$ and the price of ghee decreased by $5 \%.$ If the importance of oil is three times and of rice is double that of ghee and the importance of each of wheat and bajri is double that of rice, find price index number of the group of these five items and interpret it.
Answer
Here, percentage increase or decrease in the prices of food items are given.
$\therefore $ Index number of the item $= 100 + \%$ increase OR
$= 100 – \%$ decrease
$\therefore $ Suppose, importance of ghee is $1.$
$\therefore $ the importance of oil is $3 \times 1 = 3$ and that of rice is $2 \times 1 = 2$
The importance of wheat and bajri each is double the importance of rice,
$\therefore $ the importance of wheat is $2 \times 2 = 4$ and that of bajri is $2 \times 2 = 4$​​​​​​​

General index number of price $ I =\frac{\Sigma I W}{\Sigma W}$
$= \frac{1775}{14} = 126.79$
Hence, price index of a group of five items is obtained $126.79.$
Interpretation: There is $26.79 \%$ in the prices of five items of the group.
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Question 94 Marks
The index numbers of different groups of industrial output of a city and the weights of these groups are given below. Find the index number of the industrial production.
Answer

Index number of industrial production $I = \frac{\Sigma \mathrm{IW}}{\Sigma \mathrm{W}}$
$= \frac{37918.9}{100}$
$= 379.19$
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Question 104 Marks
Use the following information to find $(i)$ fixed base index numbers with the year $2008$ as the base year and $(ii)$ the index numbers by taking the average price of the years $2008$ and $2009$ as the base year price:
Answer
$(i)$ Base year $= 2008$

$(ii)$ Base year $=$ Average of prices of the year $2008$ and $2009:$
Average price of $2008$ and 2009 $p_0$ = $\frac{(\text { Price of } 2008)+(\text { Price of } 2009)}{2}$
$= \frac{32+38}{2}$
$= \frac{70}{2} = ₹ 35$​​​​​​​

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Question 114 Marks
Compute the index number by the method of total expenditure for the year $2014$ by taking the base year $2013$ using the following data:
Answer
Here, base year is $2013$ and current year is $2014.$
$\therefore p_0=$ Price in $2013; p_1=$ Price in $2014$ and $q_1=$ Quantity in $2014$
The table for calculation is prepared as follows:

Index number by total expenditure method = $\frac{\Sigma p_{1} q_{1}}{\Sigma p_{0} q_{1}} \times 100$
$= \frac{1180}{1020} \times 100 = 115.69$
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Question 124 Marks
Compute the index number for the year $2015$ with the base year $2010$ by the method of total expenditure using the following data:
Answer
Here, base year is $2010$ and current year is $2015.$
$\therefore p _0=$ Price in $2010; q _0=$ Quantity in $2010$ and $p _1=$ Price in $2015$
The table for calculation is prepared as follows:

Index number by total expenditure method $= \frac{\Sigma p_{1} q_{0}}{\Sigma p_{0} q_{0}} \times 100 = \frac{1016}{680} \times 100 = 149.41$
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Question 134 Marks
Find the general index number using the following data about prices of different items in the year $2012$ by taking the base year $2010.$
Answer

General Index number $= \frac{\sum\left(\frac{p_{1}}{p_{0}}\right)}{n} \times 100$
$= \frac{6.1159}{5} \times 100$
$= 122.32$
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Question 144 Marks
Find the Paasche’s and Fisher’s index numbers for the year $2015$ with the base year $2014$ using the data given below:
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Question 164 Marks
Find the Laspeyre’s, Paasche’s and Fisher’s index numbers for the year $2015$ with the base year $2014$ using the following information:
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Question 174 Marks
Find the Laspeyre’s, Paasche’s and Fisher’s index numbers for the year $2015$ with the base year $2014$ using the following information:
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Question 184 Marks
The information about six different items used in the furniture items is as follows. Find the index number for the year $2015$ with the base year $2014$ and interpret it.
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Question 194 Marks
The chain base index numbers for sales of a certain type of scooter from the year $2010$ to $2015$ are as follows. Find fixed base index numbers.
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Question 204 Marks
The fixed base index numbers of food from the month of January to October in the year $2015$ for the industrial workers of Ahmedabad are as given below. Compute the chain base index numbers.
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Question 214 Marks
Obtain the chain base index number from the fixed base index numbers given below with the year $2007 - 08$ as the base year for the wholesale prices of machines and equipments:
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Question 224 Marks
The chain base index numbers of agricultural production of a state from the year $2008$ to $2014$ are as follows. Compute the fixed base index numbers. $($Take $2007$ as base year$.)$
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Question 234 Marks
The prices of five fuel related items in the years $2012$ and $2014$ are as follows. Calculate the general index number for five fuel items by taking the year $2012$ as the base year and state the overall increase in the prices of fuel items.
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Question 244 Marks
The following data are obtained about the annual average prices of wheat, rice and sugar in the wholesale market of a city. Find the general index number for three items by fixed base method with base year $2011$ and by chain base method.
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Question 254 Marks
From the following data about the retail prices of sugar in a city, find the index numbers of price of sugar by $(1)$ Fixed base method with year $2008$ as base year, $(2)$ Chain base method and $(3)$ Taking the average price of sugar for the year $2009$ and $2010$ as the base year price.
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Question 264 Marks
The data about average daily wage of a group of workers employed in a factory in a city during the year $2008$ to $2015$ are as follows. Find the index number by $(1)$ Fixed base method $($taking base year $2008), (2)$ Chain base method and $(3)$ Fixed base method by taking average of average daily wages of the years $2011$ to $2013$ as the wage for the base year.
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Question 274 Marks
Index number of the year $2014$ is $100.$ It has been increased by $10\%$ in the year $2015$ and decreased by $6\%$ in the year $2016.$ It is increased by $20\%$ in the year $2017.$ Obtain the index numbers of these four years and convert them to chain base index numbers.
Answer
Index number $: 100,110,94, 120$
Chain base index number $: 100,110, 85.45, 127.66$
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