Applied Statistics Exercise 9.2

Dear Class 12 Samacheer Kalvi students, here are the Text Book Solutions for Exercise 9.2 Business Maths Chapter 9 Applied Statistics for your reference and study.

You can find the links to the other exercises below.

Applied Statistics Important Formulas

Solutions for Applied Statistics Exercise 9.1

Solutions for Applied Statistics Exercise 9.3

Solutions for Applied Statistics Exercise 9.4

Applied Statistics Exercise 9.2

$\text{1. Define Index Number.} \\$
$\text{``An Index number is a statistical measure of fluctuations in a}\\$
$\text{ variable arranged in the form of a series and using a base }\\$
$\text{period for making comparisons'' - Lawrence J Kalpan}\\$

$\text{2. State the uses of Index Number.}\\$
$\text{The uses of Index number are:}\\$
$\text{i. It is an important tool for the formulating}\\$
$\text{decision and management policies.}\\$
$\text{ii. It helps in studying the trends and tendencies.}\\$
$\text{iii. It determines the inflation and deflation in an economy.}\\$

$\text{3. Mention the classification of Index Number.} \\$
$\text{Index number can be classified as follows:}\\$
$\text{(i) Price Index Number : It measures the general changes in the retail}\\$
$\text{or wholesale price level of a particular or group of commodities.}\\$
$\text{(ii) Quantity Index Number: These are indices to measure the }\\$
$\text{changes in the quantity of goods manufactured in a factory.}\\$
$\text{(iii) Cost of living Index Number: These are intended to }\\$
$\text{study the effect of change in the price level on the} \\$
$\text{cost of living of different classes of people.} \\$

$\text{4. Define Laspeyre's price index number.}\\$
$\text{Laspeyre's Price Index Number is a weighted price index that measures} \\$
$\text{ the change in prices between the base year and the current year, using } \\$
$\text{the base-year quantities as weights.} \\$
$\text{Laspeyre's Price Index}=\dfrac{\Sigma (P_{1} \times Q_{0})}{\Sigma (P_{0} \times Q_{0})}\times 100 \\$
$\text{Where: }P_{0}= \text{price in the base year} \\$
$P_{1} =\text{ price in the current year}\\$
$Q_{0} = \text{quantity in the base year}\\$
$Q_{1} = \text{quantity in the current year} \\$

$\text{5. Explain Paasche's price index number.} \\$
$\text{Paasche's Price Index Number is a weighted price index that measures } \\$
$\text{the change in prices between the base year and the current year, } \\$
$\text{using the current-year quantities as weights.} \\$
$\text{Paasche's Price Index}=\dfrac{\Sigma (P_{1} \times Q_{1})}{\Sigma (P_{0} \times Q_{1})}\times 100 \\$
$\text{Where: }P_{0}= \text{price in the base year} \\$
$P_{1} =\text{ price in the current year}\\$
$Q_{0} = \text{quantity in the base year}\\$
$Q_{1} = \text{quantity in the current year} \\$

$\text{6. Write note on Fisher's price index number.}\\$
$\text{Fisher's Price Index Number is a weighted price index that is calculated} \\$
$\text{as the geometric mean of the Laspeyres and Paasche price index numbers. } \\$
$\text{It is considered one of the most reliable index numbers because it combines} \\$
$\text{ the advantages of both methods and minimizes their individual weaknesses.} \\$
$\text{Fisher's Price Index}=\sqrt{\dfrac{\Sigma (P_{1} \times Q_{0})}{\Sigma (P_{0} \times Q_{0})}\times \dfrac{\Sigma (P_{1} \times Q_{1})}{\Sigma (P_{0} \times Q_{1})}}\times 100 \\$
$\text{Where: }P_{0}= \text{price in the base year} \\$
$P_{1} =\text{ price in the current year}\\$
$Q_{0} = \text{quantity in the base year}\\$
$Q_{1} = \text{quantity in the current year} \\$

$\text{7. State the test of adequacy of index number.} \\$
$\text{There are two tests which are used to test the adequacy for}\\$
$\text{ an index number. The two tests are as follows:} \\$
$\text{(i) Time Reversal Test}\\$
$\text{(ii) Factor Reversal Test} \\$

$\text{8. Define Time Reversal Test.} \\$
$\text{It is an important test for testing the consistency of a good index number.}\\$
$\text{This test maintains time consistency by working both forward and backward } \\$
$\text{with respect to time (here time refers to base year and current year).} \\$
$\text{Symbolically the following relationship should be satisfied,} P_{01} \times P_{10}=1 \\$

$\text{9. Explain Factor Reversal Test.}\\$
$\text{This is another test for testing the consistency of a good index number.} \\$
$\text{The product of price index number and quantity index number from the base}\\$
$\text{ year to the current year should be equal to the true value ratio. That}\\$
$\text{is, the ratio between the total value of current period and total value}\\$
$\text{ of the base period is known as true value ratio. Factor Reversal Test }\\$
$\text{is given by:} P_{01}\times Q_{01}=\dfrac{\Sigma p_{1}q_{1}}{\Sigma p_{0}q_{0}} \\$

$\text{10. Define true value ratio.}\\$
$\text{The ratio between the total value of current period and total}\\$
$\text{value of the base period is known as true value ratio.}\\$

$\text{11. Discuss about Cost of Living Index Number.} \\$
$\text{Cost of Living Index Number is constructed to study the effect of changes}\\$
$\text{ in the price of goods and services of consumers for a current period as }\\$
$\text{compared with base period. The change in the cost of living index number}\\$
$\text{between any two periods means the change in income which will be necessary} \\$
$\text{to maintain the same standard of living in both the periods. Therefore the}\\$
$\text{cost of living index number measures the average increase in the cost to} \\$
$\text{maintain the same standard of life. Further, the consumption habits of people} \\$
$\text{ differ widely from class to class (rich, poor, middle class) and even with }\\$
$\text{the region. The changes in the price level affect the different classes of }\\$
$\text{people, consequently the general price index numbers fail to reflect the} \\$
$\text{effect of changes in their cost of living of different classes of people.} \\$
$\text{Therefore, cost of living index number measures the general price movement} \\$
$\text{of the commodities consumed by different classes of people. }\\$

$\text{12. Define Family Budget Method.}\\$
$\text{In this method, the weights are calculated by multiplying prices and} \\$
$\text{quantity of the base year.}\; (i.e.)\;V =p_{0}q_{0}. \text{ The formula is given by,}\\$
$\text{Cost of Living Index Number }= \dfrac{\Sigma PV}{\Sigma V} \; \text{where}\\$
$p=\dfrac{p_{1}}{p_{0}} \times 100 \; \text{is the price relative.} \\$

$\text{13. State the uses of Cost of Living Index Number.}\\$
$\text{(i) It indicates whether the real wages of workers are rising or}\\$
$\text{ falling for a given time.} \\$
$\text{(ii) It is used by the administrators for regulating dearness }\\$
$\text{allowance or grant of bonus to the workers.}\\$

$\text{14. Calculate by a suitable method, the index number of price} \\$
$\text{ from the following data:} \\$

Commodity	2002		2012
	Price	Quantity	Price	Quantity
A	10	20	16	10
B	12	34	18	42
C	15	30	20	26

Commodity	p₀	q₀	p₁	p₁	p₀q₀	p₀q₁	p₁q₀	p₁q₁
A	10	20	16	10	200	100	320	160
B	12	34	18	42	408	504	612	756
C	15	30	20	26	450	390	600	520
Total					1058	994	1532	1436

$\text{Laspeyre's Price Index Number}\\$
$p_{01}=\dfrac{p_{1}q_{0}}{p_{0}q_{0}}\times 100\\$
$=p_{01}=\dfrac{1532}{1058}\times 100 \\$
$p_{01}=144.80\\$
$\text{Paasche's Price Index Number}\\$
$p_{01}=\dfrac{p_{1}q_{1}}{p_{0}q_{1}}\times 100\\$
$=p_{01}=\dfrac{1436}{994}\times 100\\$
$p_{01}=144.47\\$
$\text{Fisher's Price Index Number}\\$
$p_{01}=\sqrt{p_{01}(Lasp.I.N) \times p_{10}(Paasc.I.N) }\\$
$p_{01}=\sqrt{144.80 \times 144.47}\\$
$p_{01}=144.63\\$
$\text{Price increase in Laspeyre's method:} \;44.80\%\\$
$\text{Price increase in Paasche's method:} \;44.47\%\\$
$\text{Price increase in Fisher's method:} \;44.63\% \\$

$\text{15. Calculate price index number for 2005 by}\\$
$\text{(a) Laspeyre's (b) Paasche's method}\\$

Commodity	1995		2005
	Price	Quantity	Price	Quantity
A	5	60	15	70
B	4	20	8	35
C	3	15	6	20

Commodity	p₀	q₀	p₁	p₁	p₀q₀	p₀q₁	p₁q₀	p₁q₁
A	5	60	15	70	300	350	900	1050
B	4	20	8	35	80	140	160	280
C	3	15	6	20	45	60	90	120
Total					425	550	1150	1450

$\text{Laspeyre's Price Index Number}\\$
$p_{01}=\dfrac{p_{1}q_{0}}{p_{0}q_{0}}\times 100\\$
$=p_{01}=\dfrac{1150}{425}\times 100 \\$
$p_{01}=270.59\\$
$\text{Paasche's Price Index Number}\\$
$p_{01}=\dfrac{p_{1}q_{1}}{p_{0}q_{1}}\times 100\\$
$=p_{01}=\dfrac{1450}{550}\times 100\\$
$p_{01}=263.64\\$
$\text{Fisher's Price Index Number}\\$
$p_{01}=\sqrt{p_{01}(Lasp.I.N) \times p_{10}(Paasc.I.N) }\\$
$p_{01}=\sqrt{144.80 \times 144.47}\\$
$p_{01}=144.63\\$
$\text{Price increase in Laspeyre's method:} \;170.59\%\\$
$\text{Price increase in Paasche's method:} \;163.64\% \\$

$\text{16. Compute (i) Laspeyre's (ii) Paasche's (iii) Fisher's Index}\\$
$\text{numbers for the 2010 from the following data.}\\$

Commodity	Price		Quantity
	2000	2010	2000	2010
A	12	14	18	16
B	15	16	20	15
C	14	15	24	20
D	12	12	29	23

Commodity	p₀	q₀	p₁	p₁	p₀q₀	p₀q₁	p₁q₀	p₁q₁
A	12	14	18	16	216	192	252	224
B	15	16	20	15	300	225	320	240
C	14	15	24	20	336	280	360	300
D	12	12	29	23	348	216	348	276
Total					1200	973	1280	1040

$\text{Laspeyre's Price Index Number}\\$
$p_{01}=\dfrac{p_{1}q_{0}}{p_{0}q_{0}}\times 100\\$
$=p_{01}=\dfrac{1280}{1200}\times 100 \\$
$p_{01}=106.67\\$
$\text{Paasche's Price Index Number}\\$
$p_{01}=\dfrac{p_{1}q_{1}}{p_{0}q_{1}}\times 100\\$
$=p_{01}=\dfrac{1040}{973}\times 100\\$
$p_{01}=106.89\\$
$\text{Fisher's Price Index Number}\\$
$p_{01}=\sqrt{p_{01}(Lasp.I.N) \times p_{10}(Paasc.I.N) }\\$
$p_{01}=\sqrt{106.67 \times 106.89}\\$
$p_{01}=106.78\\$
$\text{Price increase in Laspeyre's method:} \;6.67\%\\$
$\text{Price increase in Paasche's method:} \;6.89\%\\$
$\text{Price increase in Fisher's method:} \;6.78\% \\$

$\text{17. Using the following data, construct Fisher's Ideal index and show how}\\$
$\text{ it satisfies Factor Reversal Test and Time Reversal Test?}\\$

Commodity	Price in Rupees per unit		Number of units
	Base year	Current year	Base year	Current year
A	6	10	50	56
B	2	2	100	120
C	4	6	60	60
D	10	12	50	24
E	8	12	40	36

Commodity	p₀	q₀	p₁	p₁	p₀q₀	p₀q₁	p₁q₀	p₁q₁
A	6	10	50	56	300	336	500	560
B	2	2	100	120	200	240	200	240
C	4	6	60	60	240	240	360	360
D	10	12	50	24	500	240	600	288
E	8	12	40	36	320	288	480	432
Total					1560	1344	2140	1880

$\text{Fisher's Price Index Number}\\$
$\sqrt{\dfrac{\Sigma p_{1}q_{0}}{\Sigma p_{0}q_{0}} \times \dfrac{\Sigma p_{1}q_{1}}{\Sigma p_{0}q_{1}}} \times 100\\$
$\sqrt{\dfrac{2140}{1560} \times \dfrac{1880}{1344}} \times 100\\$
$p_{01}=138.52 \\$

$\text{Factor Reversal Test}\\$
$p_{01}\times q_{01}=\dfrac{p_{1}q_{1}}{p_{0}q_{0}}\\$
$p_{01}=\sqrt{\frac{\Sigma p_{1}q_{0}\times \Sigma p_{1}q_{1} }{\Sigma p_{0}q_{0}\times \Sigma p_{0}q_{1}}}\\$
$q_{01}=\sqrt{\frac{\Sigma q_{1}p_{0}\times \Sigma q_{1}p_{1} }{\Sigma q_{0}p_{0}\times \Sigma q_{0}p_{1}}}\\$
$p_{01}q_{01}=\sqrt{\frac{\Sigma p_{1}q_{0}\times \Sigma p_{1}q_{1} }{\Sigma p_{0}q_{0}\times \Sigma p_{0}q_{1}} \times\frac{\Sigma q_{1}p_{0}\times \Sigma q_{1}p_{1} }{\Sigma q_{0}p_{0}\times \Sigma q_{0}p_{1}}}\\$
$p_{01}q_{01}=\sqrt{\dfrac{2140 \;\times 1880\;\times1344\;\times1880}{1560\;\times1344\;\times1560\;\times2140}} \\$
$p_{01}q_{01}=\sqrt{\dfrac{1880\;\times1880}{1560\;\times1560}} \\$
$p_{01}q_{01}=\dfrac{1880}{1560}\\$
$p_{01}q_{01}=\dfrac{p_{1}q_{1}}{p_{0}q_{0}}\\$

$\text{Time Reversal Test}=p_{01} \times p_{10}=1 \\$
$p_{01}=\sqrt{\dfrac{\Sigma p_{1}q_{0}\times \Sigma p_{1}q_{1} }{\Sigma p_{0}q_{0}\times \Sigma p_{0}q_{1}}}\\$
$p_{10}=\sqrt{\dfrac{\Sigma p_{0}q_{1}\times \Sigma p_{0}q_{0} }{\Sigma p_{1}q_{1}\times \Sigma p_{1}q_{0}}}\\$
$p_{01}\times p_{10}=\sqrt{\dfrac{\Sigma p_{1}q_{0}\times \Sigma p_{1}q_{1} \times \Sigma p_{0}q_{1} \times \Sigma p_{0}q_{0}}{\Sigma p_{0}q_{0}\times \Sigma p_{0}q_{1}\times \Sigma p_{1}q_{1} \times \Sigma p_{1}q_{0} }}\\$
$p_{01}\times p_{10}=\sqrt{\dfrac{2140 \times 1880\;\times1344 \times 1560}{1560 \times 1344 \times 1880 \times 2140}}\\$
$=\sqrt{1}\\$
$p_{01}\times p_{10}=1\\$

$\text{18. Using Fisher's Ideal Formula, compute price index number for 1999}\\$
$\text{ with 1996 as base year, given the following:}\\$

Year	Commodity: A		Commodity: B		Commodity: C
	Price (Rs.)	Quantity (Kg)	Price (Rs.)	Quantity (Kg)	Price (Rs.)	Quantity (Kg)
1996	5	10	8	6	6	3
1999	4	12	7	7	5	4

Commodity	p₀	q₀	p₁	q ₁	p₀q₀	p₀q₁	p₁q₀	p₁q₁
A	5	10	4	12	50	60	40	48
B	8	6	7	7	48	56	42	49
C	6	3	5	4	18	24	15	20
Total					116	140	97	117

$\text{Fisher's Price Index Number}\\$
$\sqrt{\dfrac{\Sigma p_{1}q_{0}}{\Sigma p_{0}q_{0}}} \times \dfrac{\Sigma p_{1}q_{1}}{\Sigma p_{0}q_{1}} \times 100\\$
$\sqrt{\dfrac{97}{116}} \times \dfrac{117}{140} \times 100\\$
$p_{01}=83.60 \\$

$\text{19. Calculate Fisher's index number to the following data. Also show that it}\\$
$\text{ satisfies Time Reversal Test.}\\$

Commodity	2016		2017
	Price (Rs.)	Quantity (Kg)	Price (Rs.)	Quantity (Kg)
Food	40	12	65	14
Fuel	72	14	78	20
Clothing	36	10	36	15
Wheat	20	6	42	4
Others	46	8	52	6

Commodity	p₀	q₀	p₁	q ₁	p₀q₀	p₀q₁	p₁q₀	p₁q₁
Food	40	12	65	14	480	560	780	910
Fuel	72	14	78	20	1008	1440	1092	1560
Clothing	36	10	36	15	360	540	360	540
Wheat	20	6	42	4	120	80	252	168
Others	46	8	52	6	368	276	416	312
Total					2336	2896	2900	3490

$\text{Fisher's Price Index Number}\\$
$\sqrt{\dfrac{\Sigma p_{1}q_{0}}{\Sigma p_{0}q_{0}}} \times \dfrac{\Sigma p_{1}q_{1}}{\Sigma p_{0}q_{1}} \times 100\\$
$\sqrt{\dfrac{2900}{2336}} \times \dfrac{3490}{2896} \times 100\\$
$p_{01}=122.31\\$

$\text{Time Reversal Test}=p_{01} \times p_{10}=1 \\$
$p_{01}=\sqrt{\dfrac{\Sigma p_{1}q_{0}\times \Sigma p_{1}q_{1} }{\Sigma p_{0}q_{0}\times \Sigma p_{0}q_{1}}}\\$
$p_{10}=\sqrt{\dfrac{\Sigma p_{0}q_{1}\times \Sigma p_{0}q_{0} }{\Sigma p_{1}q_{1}\times \Sigma p_{1}q_{0}}}\\$
$p_{01}\times p_{10}=\sqrt{\dfrac{\Sigma p_{1}q_{0}\times \Sigma p_{1}q_{1} \times \Sigma p_{0}q_{1} \times \Sigma p_{0}q_{0}}{\Sigma p_{0}q_{0}\times \Sigma p_{0}q_{1}\times \Sigma p_{1}q_{1} \times \Sigma p_{1}q_{0} }}\\$
$p_{01}\times p_{10}=\sqrt{\dfrac{2900\times 3490\;\times 2896\times 2336}{2336\times 2896\times 3490\times 2900}}\\$
$p_{01}\times p_{10} =\sqrt{1}\\$
$p_{01}\times p_{10}=1\\$

$\text{20. The following are the group index numbers and the group weights of an }\\$
$\text{ average working class family's budget. Construct the cost of living index number:}\\$

Groups	Food	Fuel and Lighting	Clothing	Rent	Miscellaneous
Index Number	2450	1240	3250	3750	4190
Weight	48	20	12	15	10

Groups	P(IN)	V(W)	PV
Food	2450	48	117600
Fuel and Lighting	1240	20	24800
Clothing	3250	12	39000
Rent	3750	15	56250
Miscellaneous	4190	10	41900
Total		105	279550

$\text{Cost of Living I.N}=\dfrac{\Sigma PV}{\Sigma V}\\$
$\text{Cost of Living I.N}= =\dfrac{279550}{105}\\$
$\text{Cost of Living I.N}=2662.38\\$

$\text{21. Construct the cost of living Index number for 2015 on the basis} \\$
$\text{ of 2012 from the following data using family budget method.} \\$

Commodity	Price		Weights
	2012	2015
Rice	250	280	10
Wheat	70	85	5
Corn	150	170	6
Oil	25	35	4
Dhal	85	90	3

Commodity	p₀	p₁	V	p=p₁/p₀ x 100	PV
Rice	250	280	10	112	1120
Wheat	70	85	5	121.43	607.15
Corn	150	170	6	113.33	679.98
Oil	25	35	4	140	560
Dhal	85	90	3	105.88	317.64
Total			28		3284.77

$\text{Cost of Living I.N}=\dfrac{\Sigma PV}{\Sigma V}\\$
$\text{Cost of Living I.N}= =\dfrac{3284.77}{28}\\$
$\text{Cost of Living I.N}=117.31\\$

$\text{22. Calculate the cost of living index by aggregate expenditure method:} \\$

Commodity	Weights 2010	Price
		2010	2015
P	80	22	25
Q	30	30	45
R	25	42	50
S	40	25	35
T	50	36	52

Commodity	p₀	q₀	p₁	p₁q₀	p₁ q₀
P	22	80	25	1760	2000
Q	30	30	45	900	1350
R	42	25	50	1050	1250
S	25	40	35	1000	1400
T	36	50	52	1800	2600
Total				6510	8600

$\text{Cost of Living I.N}=\dfrac{\Sigma p_{1}q_{0}}{\Sigma p_{0}q_{0}}\times 100\\$
$\text{Cost of Living I.N}= =\dfrac{8600}{6510}\\$
$\text{Cost of Living I.N}=132.10\\$
$\text{Increased by } 32.10\% \\$

Applied Statistics Exercise 9.2

Applied Statistics Exercise 9.2

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Applied Statistics Exercise 9.2

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