Ch index number statistics grade 11

Page No 422:

Question 1:

Using Simple Aggregate Method and Price Relatives Method, find out index values for the year 2021 from the following data:

Items	A	B	C	D	E
2011 Price (â‚¹)	15	33	38	25	50
2021 Price (â‚¹)	30	35	57	35	63

ANSWER:

Simple Aggregate Method

Items	2011 Price (P0)	2021 Price (P1)
A B C D E	15 33 38 25 50	30 35 57 35 63
	ΣP0 = 161	ΣP1 = 220

P_{01} = \frac{{ΣP}_{1}}{{ΣP}_{0}} \times 100 or, P_{01} = \frac{220}{161} \times 100 \Rightarrow P_{01} = 136.64

Price Relative Method

Items	2011 Price (P0)	2021 Price (P1)	Price relative = $\frac{P_{1}}{P_{0}} \times 100$
A	15	30	$\frac{30}{15} \times 100 = 200$
B	33	35	$\frac{35}{33} \times 100 = 106.06$
C	38	57	$\frac{57}{38} \times 100 = 150$
D	25	35	$\frac{35}{25} \times 100 = 140$
E	50	63	$\frac{63}{50} \times 100 = 126$
N = 5			$Σ (\frac{P_{1}}{P_{0}} \times 100) = 722.06$

P_{01} = \frac{Σ (\frac{P_{1}}{P_{0}} \times 100)}{N} or, P_{01} = \frac{722.06}{5} \Rightarrow P_{01} = 144.412

Page No 422:

Question 2:

Find out index value by the Price Relative Method for the year 2021 from the following data:

Items	A	B	C	D	E	F	G
2011 Price (â‚¹)	100	10	5	4	1	2	3
2021 Price (â‚¹)	100	9	4	2	1	2.50	2.25

ANSWER:

Items	2011 Price (P0)	2021 Price (P1)	Price relative $= \frac{P_{1}}{P_{0}} \times 100$
A	100	100	$\frac{100}{100} \times 100 = 100$
B	10	9	$\frac{9}{10} \times 100 = 90$
C	5	4	$\frac{4}{5} \times 100 = 80$
D	4	2	$\frac{2}{4} \times 100 = 50$
E	1	1	$\frac{1}{1} \times 100 = 100$
F	2	2.50	$\frac{2.50}{2} \times 100 = 125$
G	3	2.25	$\frac{2.25}{3} \times 100 = 75$
N = 7			$Σ (\frac{P_{1}}{P_{0}} \times 100) = 620$

P_{01} = \frac{Σ (\frac{P_{1}}{P_{0}} \times 100)}{N} or, P_{01} = \frac{620}{7} = 88.57

Hence, Price Index = 88.57

Page No 422:

Question 3:

Construct an index number by Price Relatives Method using 2011 as base year:

Goods	A	B	C	D
2011 Price (â‚¹)	8	10	15	20
2020 Price (â‚¹)	10	12	18	22
2021 Price (â‚¹)	12	14	20	25

ANSWER:

Goods	2011 Price (P0)	2020 Price (P1)	2021 Price (P2)	Price relatives of 2020 in relation to 2011 $(\frac{P_{1}}{P_{0}} \times 100)$	Price relatives of 2021 in relation to 2011 $(\frac{P_{2}}{P_{0}} \times 100)$
A	8	10	12	$\frac{10}{8} \times 100 = 125$	$\frac{12}{8} \times 100 = 150$
B	10	12	14	$\frac{12}{10} \times 100 = 120$	$\frac{14}{10} \times 100 = 140$
C	15	18	20	$\frac{18}{15} \times 100 = 120$	$\frac{20}{15} \times 100 = 133.33$
D	20	22	25	$\frac{22}{20} \times 100 = 110$	$\frac{25}{20} \times 100 = 125$
N = 4				$Σ (\frac{P_{1}}{P_{0}} \times 100) = 475$	$Σ (\frac{P_{2}}{P_{0}} \times 100) = 548.33$

P_{01} = \frac{Σ (\frac{P_{1}}{P_{0}} \times 100)}{N} = \frac{475}{4} = 118.75 P_{02} = \frac{Σ (\frac{P_{2}}{P_{0}} \times 100)}{N} = \frac{548.33}{4} = 137.08

Page No 422:

Question 4:

Construct and index of prices using 2011 as the base year and Price Relatives Method:

Goods	Weight	2011	2020	2021
A B C	5 3 2	10 5 4	12 6 5	14 8 7

ANSWER:

Construction of Price Index for the year 2020

Goods	Weight (W)	2011 Price (P0)	2020 Price (P1)	$R = \frac{P_{1}}{P_{0}} \times 100$	RW
A	5	10	12	$\frac{12}{10} \times 100 = 120$	600
B	3	5	6	$\frac{6}{5} \times 100 = 120$	360
C	2	4	5	$\frac{5}{4} \times 100 = 125$	250
	ΣW = 10				ΣRW= 1210

P_{01} = \frac{Σ R W}{Σ W} or, P_{01} = \frac{1210}{10} = 121

Construction of Price Index for the year 2021

Goods	Weight (W)	2011 Price (P0)	2021 Price (P2)	$R = \frac{P_{2}}{P_{0}} \times 100$	RW
A	5	10	14	$\frac{14}{10} \times 100 = 140$	700
B	3	5	8	$\frac{8}{5} \times 100 = 160$	480
C	2	4	7	$\frac{7}{4} \times 100 = 175$	350
	ΣW = 10				ΣRW= 1530

P_{02} = \frac{Σ R W}{Σ W} or, P_{02} = \frac{1530}{10} = 153

Page No 454:

Question 1:

↵Taking 2011 as base year, construct the index numbers of the years 2017 and 2021.

Year	2011	2017	2018	2019	2020	2021
Prices (â‚¹)	10	14	16	20	22	24

ANSWER:

Year	Price
2011	10
2017	$14$
2018	16
2019	20
2020	22
2021	$24$

Since, base year is given as 2011
∴ P0 = 10
Index number for year 2017

Here, P1 = 14

P_{01} = \frac{P_{1}}{P_{0}} \times 100 = \frac{14}{10} \times 100 = 140

Index number for year 2021
Here, P1 = 24
Substituting the values in the formula

P_{01} = \frac{24}{10} \times 100 = 240

Page No 455:

Question 2:

Construct index number by Price Relative Method taking 2011 as base year:

**Price per Unit in â‚¹**
Year	A	B	C	D
2011 2019 2020 2021	25 20 25 28	18 22 20 24	16 24 25 30	21 22 25 26

ANSWER:

Here, we construct the index number for each of the years from 2019-2021.
Base Year 2011, Current year 2019

	2011 ( P0)	2019 (P1)	Price Relative = $\frac{P_{1}}{P_{0}} \times 100$
A	25	20	$\frac{20}{25} \times 100 = 80$
B	18	22	$\frac{22}{18} \times 100 = 122.22$
C	16	24	$\frac{24}{16} \times 100 = 150$
D	21	22	$\frac{22}{21} \times 100 = 104.76$
			∑ =456.96

According to the Price Relative Method, price index is calculated using the following formula.

P_{01} = \frac{Σ (\frac{P_{1}}{P_{0}} \times 100)}{N} Substituting the values in the formula . P_{01} = \frac{456.98}{4} = 114.245

P_{01} = \frac{456.98}{4} = 114.245

Base Year 2011, Current year 2020

	2011 (P0)	2020 (P1)	Price Relative = $\frac{P_{1}}{P_{0}} \times 100$
A	25	25	$\frac{25}{25} \times 100 = 100$
B	18	20	$\frac{20}{18} \times 100 = 111.11$
C	16	25	$\frac{25}{16} \times 100 = 156.25$
D	21	25	$\frac{25}{21} \times 100 = 119.04$
			∑= 486.4

P_{01} = \frac{486.4}{4} = 121.60

Base Year 2011, Current Year 2021

	2011 (P0)	2021 (P1)	Price Relative = $\frac{P_{1}}{P_{0}} \times 100$
A	25	28	$\frac{28}{25} \times 100 = 280$
B	18	24	$\frac{24}{18} \times 100 = 133.33$
C	16	30	$\frac{30}{16} \times 100 = 187.5$
D	21	26	$\frac{26}{21} \times 100 = 123.80$
			∑ 556.63

P_{01} = \frac{556.63}{4} = 139.157

Page No 455:

Question 3:

Compute a Price Index for the following by (i) Simple Aggregative Method, and (ii) Average of Price Relative Method:

Commodities	A	B	C	D	E	F
Price in 2011 (â‚¹)	20	30	10	25	40	50
Price in 2021 (â‚¹)	25	30	15	35	45	55

ANSWER:

(i) Simple Aggregate Method

	2011 (P0)	2021 (P1)
A B C D E F	20 30 10 25 40 50	25 30 15 35 45 55





	∑P0=â€‹ 175	∑P1 = 205

P_{01} = \frac{Σ P_{1}}{Σ P_{0}} = \frac{205}{175} \times 100 = 117.14

(ii) Price Relative Method

	2011 â€‹(P0)	2021 â€‹(P1)	Price Relative = $\frac{P_{0}}{P_{1}} \times 100$
A	20	25	$\frac{25}{20} \times 100 = 125$
B	30	30	$\frac{30}{30} \times 100 = 100$
C	10	15	$\frac{15}{10} \times 100 = 150$
D	25	35	$\frac{35}{25} \times 100 = 140$
E	40	45	$\frac{45}{40} \times 100 = 112.5$
F	50	55	$\frac{55}{50} \times 100 = 110$
			∑ = 737.5

P_{01} = \frac{Σ (\frac{P_{0}}{P_{1}} \times 100)}{N} = \frac{737.5}{6} = 122.91

Page No 455:

Question 4:

Construct price index number of the following data by using:
(i) Laspeyre's Method, (ii) Paasche's Method, and (iii) Fisher's Method.

Items	Base Year		Current Year
Items	Quantity	Price	Quantity	Price
A B C D	3 7 4 6	5 4 7 6	2 5 3 5	8 6 10 7

ANSWER:

	q0	p0	p0q0	p1	q1	p1q1	p1q0	p0q1
A B C D	3 7 4 6	5 4 7 6	15 28 28 36	8 6 10 7	2 5 3 5	16 30 30 35	24 42 40 42	10 20 21 30



			$Σ p_{0} q_{0} = 107$			$Σ p_{1} q_{1} = 111$	$Σ p_{1} q_{0} = 148$	$Σ p_{0} q_{1} = 81$

(i) Laspeyre's Price index:

P_{01} = \frac{Σ P_{1} q_{0}}{Σ P_{0} q_{0}} \times 100 o r, P_{01} = \frac{148}{107} \times 100 = 138.31

(ii) Paasche's Price index:

P_{01} = \frac{Σ P_{1} q_{1}}{Σ P_{0} q_{1}} o r, P_{01} = \frac{111}{81} \times 100 = 137.04

(iii) Fisher's Price index:

P_{01} = \sqrt{\frac{Σ P_{1} q_{0}}{Σ P_{0} q_{0}} \times \frac{Σ P_{1} q_{1}}{Σ P_{0} q_{1}}} \times 100 o r, P_{01} = \sqrt{\frac{148}{107} \times \frac{111}{81}} \times 100 = 137.67

Page No 455:

Question 5:

Construct an index number for the year 2021, taking 2011 as base year by any method you deem ideal:

Year	Good I		Good II		Good III
Year	Price	Quantity	Price	Quantity	Price	Quantity
2011 2021	5 4	10 12	8 7	6 7	6 5	3 4

ANSWER:

	P0	q0	P1	q1	P0q0	P1q1	P0q1	P1q0
I II III	5 8 6	10 6 3	4 7 5	12 7 4	50 48 18	48 49 20	60 56 24	40 42 15


					$Σ P_{0} q_{0} = 116$	$Σ P_{1} q_{1} = 117$	$Σ P_{0} q_{1} = 140$	$Σ P_{1} q_{0} = 97$

Fisher's Method

P_{01} = \sqrt{\frac{Σ P_{1} q_{0}}{Σ P_{0} q_{0}} \times \frac{Σ P_{1} q_{1}}{Σ P_{0} q_{0}}} \times 100 o r, P_{01} = \sqrt{\frac{97}{116} \times \frac{117}{140}} \times 100 = 83.59

Page No 455:

Question 6:

Given the following data and taking 2011 as the base year, construct index of prices using:

(i) Laspeyre's Method, (ii) Paasche's Method, and (iii) Fisher's Method.

Year	Commodities
	A		B		C		D
	Price	Quantity	Price	Quantity	Price	Quantity	Price	Quantity
2011 2021	24 30	8 10	9 10	3 4	16 20	5 8	10 9	3 4

ANSWER:

	P0	q0	P0q0	P1	q1	P1q1	P1q0	P0q1
A B C D	24 9 16 10	8 3 5 3	192 27 80 30	30 10 20 9	10 4 8 4	300 40 160 36	240 30 100 27	240 36 128 40



			∑P0q0 = 329			∑P1q1 = 536	∑P1q0 =397	∑P0q1â€‹ = 444

Laspeyre's Price index

P_{01} = \frac{Σ P_{1} q_{0}}{Σ P_{0} q_{0}} \times 100 o r, P_{01} = \frac{397}{329} \times 100 = 120.66

Paasche's Price index

P_{01} = \frac{Σ P_{1} q_{1}}{Σ P_{0} q_{1}} \times 100 o r, P_{01} = \frac{536}{444} \times 100 = 120.72

Fisher's Price index

P_{01} = \sqrt{\frac{Σ P_{1} q_{0}}{Σ P_{0} q_{0}} \times \frac{Σ P_{1} q_{1}}{Σ P_{0} q_{0}}} \times 100 o r, P_{01} = \sqrt{\frac{397}{329} \times \frac{536}{329}} \times 100 = 120.65

Page No 456:

Question 7:

Construct a weighted index number of the following data using price relative method:

Item	A	B	C	D	E
Base Year (Quantity)	24	14	8	4	8
Base Year (Price)	2	4	6	10	5
Current Year (Price)	3	5	9	12	5

ANSWER:

	P0	q0	P1	P0q0 â€‹(W)	$R = \frac{P_{1}}{P_{0}} \times 100$	RW
A	2	24	3	48	$\frac{3}{2} \times 100 = 150$	7200
B	4	14	5	56	$\frac{5}{4} \times 100 = 125$	7000
C	6	8	9	48	$\frac{9}{6} \times 100 = 150$	7200
D	10	4	12	40	$\frac{12}{10} \times 100 = 120$	4800
E	5	8	5	40	$\frac{5}{5} \times 100 = 100$	4000
				$Σ W = 232$		$Σ R W = 30, 200$

Weighted index number

P_{01} = \frac{Σ R W}{Σ W} \frac{30, 200}{232} o r, P

01 =130.17

Page No 456:

Question 8:

Find out the index number of the following data with Laspeyre's Method:

Commodity	2017		2018
Commodity	Price	Quantity	Price	Quantity
A B	70 62	7 3	80 74	6 2

ANSWER:

	P0	q0	P0q0	P1	q1	P1q0
A B	70 62	7 3	490 186	80 74	6 2	560 222

			∑ P0q0=676			∑P1q0= 782

Laspeyre's Price index

P_{01} = \frac{Σ P_{1} q_{0}}{Σ P_{0} q_{0}} \times 100 o r, P_{01} = \frac{782}{676} \times 100 = 115.68

Page No 456:

Question 9:

Construct Index number of the following data with Laspeyre's and Paasche's Methods:

Commodity	Base Year		Current Year
Commodity	Quantity	Price	Quantity	Price
A B C	10 8 5	0.80 0.85 1.30	11 9 5.5	0.70 0.90 0.80

ANSWER:

	P0	q0	P0q0	P1	q1	P1q1	P0q1	P1q0
A B C	0.8 0.85 1.35	10 8 5	8 6.8 6.5	0.7 0.9 0.8	11 9 5.5	7.7 8.1 4.4	8.8 7.65 7.15	7 7.2 4


			$Σ P_{0} q_{0} = 21.3$			$Σ P_{1} q_{1} = 20.2$	$Σ P_{0} q_{1} = 23.60$	$Σ P_{1} q_{0} = 18.2$

Laspeyre's Price Index

P_{01} = \frac{Σ P_{1} q_{0}}{Σ P_{0} q_{0}} o r, P_{01} = \frac{18.2}{21.3} \times 100 = 85.44

Passche's Price index

P_{01} = \frac{Σ P_{1} q_{1}}{Σ P_{0} q_{1}} o r, P_{01} = \frac{20.2}{23.60} \times 100 = 85.59

Page No 456:

Question 10:

Construct index numbers of the following data by Fisher's Method:

Commodity	Base Year		Current Year
Commodity	Price	Value	Price	Value
A B C D E	3 5 6 4 8	18 35 42 32 24	7 10 11 6 9	10 100 55 60 36

ANSWER:

	P0	Value (Base Year)	$q_{0} = \frac{V a l u e}{P_{0}}$	P0q0	P1	Value (Current Year)	$q_{1} = \frac{Value}{P_{1}}$	P1q1	P1q0	P0q1
A B C D E	3 5 6 4 8	18 35 42 32 24	6 7 7 8 3	18 35 42 32 24	7 10 11 6 9	14 100 55 60 36	2 10 5 10 4	14 100 55 60 36	42 70 77 48 27	6 50 30 40 32




		151		$Σ P_{0} q_{0} = 151$				$Σ P_{1} q_{1} = 265$	$Σ P_{1} q_{0} = 264$	$Σ P_{0} q_{1} = 158$

Fisher's Price Index

P_{01} = \sqrt{\frac{Σ P_{1} q_{0}}{Σ P_{0} q_{0}} \times \frac{Σ P_{1} q_{1}}{Σ P_{0} q_{0}}} \times 100 o r, P_{01} = \sqrt{\frac{264}{151} \times \frac{265}{158}} \times 100 = 171.24

Page No 456:

Question 11:

Construct Cost of Living Index on the basis of the following data:

Items	Price	Weight
Wheat Rice Maida Pulses Oil	241 150 200 170 125	10 4 2 2 2

ANSWER:

Items	Price (P)	Weights (W)	PW
Wheat Rice Maida Pulses Oil	241 150 200 170 125	10 4 2 2 2	2410 600 400 340 250




		∑W=20	∑PW = 4000

Cost of living Index

= \frac{\sum P W}{\sum W} = \frac{4000}{620} = 200

Page No 457:

Question 12:

Construct Consumer Price Index Number with the help of the following data:

Consumer Items	Price	Weight
Food Fuel Cloth House Rent Miscellaneous	125 120 66.67 120 150	40 10 25 15 10

ANSWER:

Items	Price (P)	Weights (W)	PW
Food Fuel Cloth House rent Miscellaneous	125 120 66.67 120 150	40 10 25 15 10	5000 1200 16 66.75 1800 1500




		$Σ W = 100$	$Σ W = 11166.75$

Consumer price index

= \frac{\sum P W}{\sum W} = \frac{111 66.75}{100} = 111.67

Page No 457:

Question 13:

From the following data find Consumer Price Index or Cost of Living Index:

Items	Quantity Consumed in Current Year	Price in Base year	Price in Current Year
Rice Pulses Oil Clothing Housing Miscellaneous	30 qt 36 kg 24 l 72 metres per month per month	12 0.4 1.5 0.75 20 20	25 0.6 2.2 10 30 15

ANSWER:

Items	q1	P0	P1	P1q1	P0q1
Rice Pulses Oil Clothing Housing Miscellaneous	30 36 24 72 1 1	12 0.4 1.5 0.75 20 20	25 0.6 2.2 10 30 15	750 21.6 52.8 720 30 15	360 14.4 36 54 20 20
				$Σ P_{1} q_{1} = 1589.4$	$Σ P_{0} q_{1} = 504.4$

â€‹Note that here we are given the current year quantities for different items. So, the Consumer Price Index is calculated using the following formula.

CPI = \frac{Σ P_{1} q_{1}}{Σ P_{0} q_{1}} or, CPI = \frac{1589.4}{504.4} \times 100 = 315.10

Page No 457:

Question 14:

Construct Cost of Living Index Number for the year 2021 from the following statistics:

Commodity	2011 Price	2011 Quantity	2021 Price
A B C D E	25 36 12 6 28	16.0 7.0 3.5 2.5 4.0	35 48 16 10 28

ANSWER:

	P0	q0	P1	P1q0	P0q0
A B C D E	25 36 12 6 28	16 7 3.5 2.5 4	35 48 16 10 28	560 336 56 25 112	400 252 42 15 112
				$Σ P_{1} q_{0} = 821$	$Σ P_{0} q_{0} = 1089$

Cost of Living Index

CPI = \frac{Σ P_{0} q_{0}}{Σ P_{1} q_{0}} \times 100 or, CPI = \frac{1089}{821} \times 100 = 132.643

Page No 457:

Question 15:

Find the Consumer Price Index from the following data. Using
(i) Aggregative Expenditure Method, and
(ii) Family Budget Method.
Is there any difference between the two results?

Commodity	Quantity Consumed in the year 2011	Unit	Price in 2011 (â‚¹)	Price in 2021 (â‚¹)
Rice Wheat Bajra Arhar Desi Ghee Sugar	6 8 1 2 20 1	Quintal Quintal Quintal Quintal kg Quintal	100 80 70 120 12 160	120 90 70 115 15 170

ANSWER:

	q0	P0	P1	W = P0q0	P1q0	$R = \frac{P_{1}}{P_{0}} \times 100$	WR
Rice	6	100	120	600	720	$\frac{120}{100} \times 100 = 120$	72,000
Wheat	8	80	90	640	720	$\frac{90}{80} \times 100 = 112.5$	72,000
Bajra	1	70	70	70	70	$\frac{70}{70} \times 100 = 100$	70,000
Arhar	2	120	115	240	230	$\frac{115}{120} \times 100 = 95.83$	22,999.2
Ghee	20	12	15	240	300	$\frac{15}{12} \times 100 = 125$	30,000
Sugar	1	160	170	160	170	$\frac{170}{160} \times 100 = 106.25$	17,000
				$Σ P_{0} q_{0} = 1950$	$Σ P_{1} q_{0} = 2210$		283999.2

(i) Aggregate expenditure methodâ€‹

C P I = \frac{Σ P_{1} q_{0}}{Σ P_{0} q_{0}} or, C P I = \frac{2210}{1950} \times 100 = 113.33

(ii) Family Budged Method

C P I = \frac{\sum W R}{\sum W} = \frac{283999.2}{1950} = 145.64

Yes, there is a difference between the values calculated by the two methods.
Difference = 145.64 – 113.33 = 32.31

Page No 458:

Question 16:

Construct index number of industrial production in the year 2021 from the following data on the basis of 2011's production:

Industry	Units	2011	2021	Weight
Electrical and Electronics Metallurgical Mechanical Mining Textiles Miscellaneous	Mill. Nos. Th. Tonnes Th. Tonnes Th. Tonnes Mill. Mtrs. Th. Tonnes	12 22 72 100 60 123	70 37 105 123 130 270	36 12 10 22 8 12

ANSWER:

Industry	q0	q1	Weight (W)	$R = \frac{q_{1}}{q_{0}} \times 100$	WR
Electronics	12	70	36	$\frac{70}{12} \times 100 = 583.33$	20999.88
Metallurgical	22	37	12	$\frac{37}{22} \times 100 = 168.18$	2018.16
Mechanical	72	105	10	$\frac{105}{72} \times 100 = 145.83$	1458.3
Mining	100	123	22	$\frac{123}{100} \times 100 = 123$	2706
Textiles	60	130	8	$\frac{130}{60} \times 100 = 216.67$	1733.36
Misce.	123	270	12	$\frac{270}{123} \times 100 = 219.51$	2634.12
			∑W = 100		∑ WR = 31549.82

Index number of Industrial Production

IIP = \frac{\sum W R}{\sum W} or, IIP = \frac{31549.82}{100} = 315.50