B.Sc. II Practical -III Expt.- 07 Title: INDEX NUMBER-I (Computations of Index Numbers by different methods)

P. V. P. Mahavidyalaya, Kavathe Mahankal

DEPARTMENT OF STATISTICS

Expt.- 07 B. Sc. II / Practical-III Date: 17/08/2023

Title: INDEX NUMBER-I (Computations of Index Numbers by different methods)

Q. 1. Obtain price index number by average of price relative’s method using average as:

(i) Arithmetic mean and (ii) Geometric mean.

Commodity	A	B	C	D
Price in Rs ( Base Year)	100	80	40	10
Price in Rs. ( Current Year)	150	100	50	20

Q. 2. Compute price index number for the following data by applying weighted average of price relatives by using average as:

(i) Arithmetic mean and (ii) Geometric mean.

Commodity	Price in 2015(Rs.)	Price in 2020 (Rs.)	Weight
Wheat	22.00	32.50	35
Sugar	23.00	33.25	20
Milk	21.50	31.75	15

Q. 3. Consider the following prices and quantities of four commodities A, B, C and D for base and current year:

Commodity	Base Year		Current Year
Commodity	Price	Quantity	Price	Quantity
A	8	50	10	60
B	10	40	12	50
C	5	100	8	120
D	6	300	9	250

Compute: i) Laspeyre’s, Paasche’s and Fisher’s price and quantity index number.

ii) Value index number. Comment on values of each of the above indices.

Q.4. Construct the price Index Number by simple aggregate method.

Items	A	B	C	D
Price in 1965 ( Po)	50	100	30	40
Price in 1971 ( P1)	65	105	50	40

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Aim: – Computation and interpretation of values of various index numbers. Testing various tests of Index Numbers.

Statistical Tools: – Simple and weighted index numbers. Various Tests of Index Numbers.

Required Formulae:–

Results:-

Price index number for given data by average of price relatives method by using average as:

(i) Arithmetic mean is equal to 150 and (ii) Geometric mean is equal to 147.14

Price index number for given data by weighted average of price relatives method

(i) By using average as Arithmetic mean is equal to 146.81

(ii) By using average as Geometric mean is equal to 146.80

Results:- For given data set:

i)   Laspeyre’s price and quantity index numbers are respectively, 144.5161 and 99.3548.

Therefore, we conclude that prices of commodities are increased by around 45% but quantities (consumption) of commodities are decreased by around 1% in current year than that of base year.

ii)   Paasche’s price and quantity index numbers are respectively, 143.1818 and 98.4375.

Therefore, we conclude that prices of commodities are increased by around 43% but quantities (consumption) of commodities are decreased by around 2% in current year than that of base year.

iii)   Fisher’s price and quantity index numbers are respectively, 143.8474 and 98.8951.

Therefore, we conclude that prices of commodities are increased by around 44% but quantities (consumption) of commodities are decreased by around 1% in current year than that of base year.

iv) Value index number is equal to 142.2581.

Therefore, we conclude that value of commodities is increased by around 42% in current year than that of base year.

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P. V. P. Mahavidyalaya, Kavathe Mahankal

DEPARTMENT OF STATISTICS

Expt.- 08 B. Sc. II / Practical-III Date: 24 /08 /2023

Title: INDEX NUMBER-II (Cost of living index number and tests of index numbers)

Q. 1. Consider the Data given in above problem Q.3 and discuss about time reversal test for price indices and factor reversal test for each of the followings:
i) Laspeyre’s index number ii) Paasche’s index number   iii) Fisher’s index number

Q.2. Following are the wage index numbers (I.N.) of industrial workers in two   companies. Compare the two series and comment.

Year

1990

1991

1992

1993

1994

1995

I.N. of Company A with base year 1992

60

90

100

140

160

200

I.N. of Company B with base year 1985

210

230

250

300

250

400

Q.3. Calculate a cost of living index number for the following two data sets by using appropriate formula and comment on them:

Q:-4 The following data relate to the income of the people & the general price index numbers of prices of a certain region. Calculate i) Real Income ii) Index number of Real Income with 1968 as the base.

Year

1968

1969

1970

1971

1972

1973

1974

Income in Rs.

800

819

825

876

920

938

924

G. P. Index No.

100

105

110

120

125

140

140

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Solution to Problem No. 1:

Results:-

i) By definition of Time Reversal Test and by above computations we conclude that only Fisher’s Price indices satisfies time reversal test.

ii) By definition of Factor Test and by above computations we conclude that only Fisher’s Index Number satisfies Factor test.

Solution to Problem No. 2:

Here first we will obtain indices for both companies A and B with common base year 1992:

Year	1990	1991	1992	1993	1994	1995
I.N. of Company A with base year 1992	60	90	100	140	160	200
I.N. of Company B with base year 1985	210	230	250	300	250	400
I.N. of Company B with base year 1992	84	92	100	120	100	160

Results:-

Observing wage indices for both companies with common base year 1992 from above table , we conclude that wages of employees in company B are more than that of A up to year 1991, but from the year 1993 wages of employees in company A are more than that of B.

Solution to Problem No. 3:

Data Set I

Commodity	Base Year		Current Year
Commodity	(p₀)	(q₀)	(p₁)	(q₁)	q₀ p₀	q₀p₁
A	5	50	7	50	250	350
B	12	10	15	100	120	150
C	8	30	10	30	240	300
D	4	40	8	4	160	320
E	9	10	12	10	90	120
					860	1240

Therefore, Cost of living index no. = 1240/860 = 144.186

Data Set II

Items	Group Index Number (I)	Weight (w)	wI
Food	350	50	17500
Fuel and Lighting	200	10	2000
Clothing	240	10	2400
House Rent	160	10	1600
Miscellaneous	250	20	5000
	Total:-	100	28500

Therefore, Cost of living index no. = 28500/100 = 285

Results:

i) Cost of Living Index Number or Consumer Price Index Number (CPI) for Data Set I is equal to 144.186.This number is greater than 100 by 44, therefore we conclude that cost in current year is increased and to maintain same living standard as that of base year in current year, it requires to spend 44% more money in current year with respect to base year. It indicates that purchasing power of money is decreased by 44% in current with respect to base year.

ii) Cost of Living Index Number or Consumer Price Index Number (CPI) for Data Set II is equal to 285.This number is greater than 100 by 185, therefore we conclude that cost in current year is increased and to maintain same living standard as that of base year in current year, it requires to spend 185% more money in current year with respect to base year. It indicates that purchasing power of money is decreased by 185% in current with respect to base year.

Solution to Problem No. 4:

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P.V.P. MAHAVIDYALAYA, KAVATHE MAHANKAL

DEPARTMENT OF STATISTICS

Expt. No.:1 B. Sc. II / Practical-III Date:

Title: Multiple Regressions

Q1. Consider the following variance co-variance matrix between X₁, X₂ and X₃:

1177	1035	444
𝑆 = [1035	4537	1501]
444	1501	787

If 𝑋¯1 = 150.0909, 𝑋¯2= 62.4546 and 𝑋¯3= 195.0 then obtain multiple regression equation of X₁ on X₂ and X₃.

Q2. Consider the following co-relation matrix (R) between X₁, X₂ and X₃:

	1	0.4479	0.4611
𝑅 =	[0.4479	1	0.7944]
	0.4611	0.7944	1

If 𝑋¯1 = 150.0909, 𝑋¯2= 62.4546, 𝑋¯3= 195.0, σ₁ = 34.3074, σ₂ = 67.3573 and σ₃ = 28.0535

then obtain multiple regression equation of X₂ on X₁ and X₃.

Q3. Consider the data on systolic blood pressure (X₁), age in years (X₂) and weight in pound (X₃):

Observation No.	1	2	3	4	5	6	7	8	9	10	11
X₁	132	143	153	162	154	168	137	159	128	166	149
X₂	52	59	67	73	64	74	54	65	46	72	61
X₃	173	184	194	211	196	220	188	207	167	217	188

Fit the linear regression equation of systolic blood pressure on age and weight. Hence estimate systolic blood pressure when age and weight of a patient are 60 years and 200 pounds, respectively.

Q4. Following is the data on phosphate adsorption index ( Y ), amount of extractable iron (X₁) and amount of extractable aluminum (X₂):

Observation No.	1	2	3	4	5	6	7	8	9	10	11	12	13
Y	4	18	14	18	26	26	21	30	28	36	65	62	40
X₁	61	175	111	124	130	178	169	169	160	244	257	333	199
X₂	13	21	24	23	64	38	33	61	39	71	112	88	54

(a) Fit the linear regression equation of phosphate adsorption index on amount of extractable iron and amount of extractable aluminum. Hence estimate phosphate adsorption index when amount of extractable iron and amount of extractable aluminum are 250 pounds and 100 pounds, respectively.

(b) Obtain residual of Y due to X₁ and X₂ for each observation and show that its sum is zero.

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Procedure of Writing Answer to Practical

Aim : Obtaining linear multiple regression equations between three variables and estimating value of dependent variable when values of independent variables are given.

Statistical Tools : Linear multiple regression equations by least square method. Required Formulae : Let (X1k, X2k, X3k), k = 1, 2, ..., N be N observations on quantitative variables (X₁, X₂, X₃).

The linear regression equation ( plane ) of X₁ on X₂ and X₃ by least square method is given by :

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DEPARTMENT OF STATISTICS

Expt.- 10 B. Sc. II / Practical-III Date: 27 /10 /2023

Title: National Income

Q 1 : From the following data determine GDP at factor cost.
a) GNP at factor cost=15000 crores b) income at factor cost from foreign nationals, residing in the country=5000 crores c)income(net) from abroad=1000 crores.

Q 2 : Given that GNP at market price is 50000 crores the total of subsidies is 5000 crores and indirect taxes paid total up to 7000 crores compute GNP at factor cost.

Q 3 : If GDP is 90000/- crores and depreciation is 2% of GDP find Net Domestic Product (NDP).

Q 4. Compute NNP at factor cost given that NDP at factor cost= 24000Crores, imports= 6000 crores and Exports= 4000 crores.

Q 5 : Determine NNP at factor cost given the following GNP at market price = 100000/- crores, Indirect taxes =7000 Crores, Subsidies =200 crores and depreciation=9000 crores. Further find NDP if net income from abroad is 2000 Crores.

Q 6 : Compute GNP, NNP and NDP given GDP= 24000 crores, net income from abroad= 3000 crores, net Exports= 1000 crores, depreciation= 500 crores.

Q 7 : From data given below find GNP at factor cost, NNP at Factor cost, NDP at factor cost, NDP at market price GNP at market price= 97000 crores, net factor income from abroad=- 200 crores, capital consumption allowance(depreciation)= 6000 crores, net indirect taxes= 10000 crores.

***

Procedure of Writing Answer to Practical

Aim: – Computation of various terms related to National Income.

Statistical Tools: – Various formulae related to National Income.

Required Formulae:–

Let GDP = Gross National Product; GNP = Gross National Product; NDP = Net Domestic Product; NNP = Net National Product;

1. GNP (Factor Cost) = GDP (Factor Cost) + Net Factor Income from Abroad – Income at factor cost from foreign Nationals

2. GNP (Factor cost) = GNP (Market price) + Subsidies – Taxes

3. NDP = GDP – Depreciation

4. NNP (Factor Cost) = NDP (Factor Cost) + X – M ,

where X = Exports and M = Imports.

5. NNP( Factor cost) = GNP( Factor cost) – Depreciation

6. GNP( Factor cost) = GNP( Market price) + Subsidies – Indirect Taxes

7. NDP = NNP – Net Income from Abroad

8. GNP = GDP + Net Income from Abroad + Exports

9. NNP = GDP + Net Income from Abroad + Exports – Depreciation

10. NNP (Factor cost) = GNP (Market price) – Taxes – Depreciation

11. NDP (Factor cost) = NNP (Factor cost) + Import – Export – Net Income from Abroad

12. NNP (Market Price) = NNP (Factor Cost) + Net Indirect tax

Data Source: – Provided in Practical Sheet for every problem.

Computations and Results: –

Solution to Problem No. P1:

We know,

GNP (Factor Cost) = GDP (Factor Cost) + Net Factor Income from Abroad – Income at Factor Cost from Foreign Nationals

Substituting given values in above formula: 15000 = GDP (Factor Cost) + 1000 – 5000

⸫ GDP (Factor Cost) = 15000 + 4000 = 19000

Result:-

GDP at Factor Cost = 19000

Solution to Problem No. P2:

We know,

GNP (Factor cost) = GNP (Market price) + Subsidies – Taxes Substituting given values in above formula:

GNP (Factor cost) = 50000 + 5000 – 7000

⸫ GNP (Factor cost) = 48000

Result:-

GNP at factor cost is equal to 48000 crores.

Solution to Problem No. P3:

We know,

NDP = GDP – Depreciation

Given: GDP = 90000 and depreciation is 2% of GDP = ⁹⁰⁰⁰⁰ × 2 = 1800

100

⸫ NDP = 90000 – 1800 = 88200

Results:-

Net Domestic Product is equal to 88200 crores

Solution to Problem No. P4:

We know,

NNP (Factor Cost) = NDP (Factor Cost) + X – M , where X = Exports and M = Imports.

Given: NNP = NDP (Factor Cost) = 24000 Crore, X = 4000 Crore and M = 6000 Crore.

⸫ NNP (Factor Cost) = 24000 + 4000 – 6000 = 22000

Results:-

NNP at factor cost is equal to 22000 Crore

Solution to Problem No. P5:

We know,

NNP( Factor cost) = GNP( Factor cost) – Depreciation

GNP( Factor cost) = GNP( Market price) + Subsidies – Indirect Taxes and

NDP = NNP – Net Income from Abroad Given:

GNP ( Market price ) = 100000/- crores, Indirect Taxes = 7000 Crores, Subsidies = 200 crores and Depreciation = 9000 crores and

Net Income from Abroad = 2000 Crores.

⸫ GNP( Factor cost) = 100000 + 200 – 7000 = 93200

NNP( Factor cost) = GNP( Factor cost) – Depreciation

= 93200 – 9000 = 84200

NDP = 84200 – 2000 = 80200

Results:-

i) NNP at factor cost is equal to 84200/- Crores ii) NDP is equal to 80200/- Crores

Solution to Problem No. P6:

We know,

GNP = GDP + Net Income from Abroad + Exports

NNP = GDP + Net Income from Abroad + Exports – Depreciation

NDP = GDP – Depreciation

Given:

GDP= 24000 crores, Net Income from Abroad = 3000 crores, Net Exports = 1000 crores and Depreciation = 500 crores.

Substituting given values in above formulae:

GNP = 24000 + 3000 + 1000 = 28000

NNP = 24000 + 3000 + 1000 – 500 = 27500

NDP = 24000 – 500 = 23500

Results:-

i) GNP is equal to 28000/- Crores

ii) NNP is equal to 27500/- Crores

iii) NDP is equal to 23500/- Crores

Solution to Problem No. P7:

We know,

GNP (Factor cost) = GNP (market price) + Subsidies – Taxes NNP (Factor cost) = GNP (market price) – Taxes – Depreciation

NDP (Factor cost) = NNP (Factor cost) + Import – Export – Net Income from Abroad NNP (Market Price) = NNP (Factor Cost) + Net Indirect tax

Given:

GNP at market price = 97000 crores, Net factor Income from Abroad =- 200 crores, Capital consumption allowance (Depreciation) = 6000 crores and

Net Indirect Taxes = 10000 crores.

⸫

GNP (Factor cost) = 97000 + 0 – 10000 = 87000 (assuming no subsidy) NNP (Factor cost) = 97000 – 10000 – 6000 = 81000

NDP (Factor cost) = 81000 + Import – Export – (–200)

= 81000 + 200 = 81200 (assuming Import = Export) NNP (Market Price) = NNP (Factor Cost) + Net Indirect tax

= 81000 + 10000 = 91000

Results:-

i) GNP (Factor cost) = 87000/- Crores . ii) NNP (Factor cost) = 81000/- Crores .

iii) NDP (Factor cost) = 81200 /- Crores . iv) NNP (Market Price) = 91000/- Crores .

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