Given below are the data relating to the production of sugarcane in a district.
Fit a straight line trend by the method of least squares and tabulate the trend values.

Year	2000	2001	2002	2003	2004	2005	2006
Prod.of Sugarcane	40	45	46	42	47	50	46

Given below are the data relating to the sales of a product in a district.
Fit a straight line trend by the method of least squares and tabulate the trend values.

Year	1995	1996	1997	1998	1999	2000	2001	2002
Sales	6.7	5.3	4.3	6.1	5.6	7.9	5.8	6.1

Calculate the seasonal index for the monthly sales of a product using the method of simple averages.

Months	Jan	Feb	Mar	Apr	May	June	July	Aug	Sep	Oct	Nov	Dec
Year
2001	15	41	25	31	29	47	41	19	35	38	40	30
2002	20	21	27	19	17	25	29	31	35	39	30	44
2003	18	16	20	28	24	25	30	34	30	38	37	39

Calculate the seasonal index for the quarterly production of a product using the method of simple averages.

Year	I Quarter	II Quarter	III Quarter	IV Quarter
2005	255	351	425	400
2006	269	310	396	410
2007	291	332	358	395
2008	198	289	310	357
2009	200	290	331	359
2010	250	300	350	400

the Laspeyre’s, Paasche’s and Fisher’s price index number for the following data. Interpret on the data.

Commodities	Price		Quandity
	2000	2010	2000	2010
Rice	38	35	6	7
Wheat	12	18	7	10
Rent	10	15	10	15
Fuel	25	30	12	16
Miscellaneous	30	33	8	10

Construct the Laspeyre’s, Paasche’s and Fisher’s price index number for the following data. Comment on the result.

Commodities	Base Year		Current Year
	Price	Quantity	Price	Quantity
Rice	15	5	16	8
Wheat	10	6	18	9
Rent	8	7	15	8
Fuel	9	5	12	6
Transport	11	4	11	7
Miscellaneous	16	6	15	10

Calculate Fisher’s price index number and show that it satisfies both Time Reversal Test and Factor Reversal Test for data given below.

Commodities	Price		Quandity
	2003	2009	2003	2009
Rice	10	13	4	6
Wheat	125	18	7	8
Rent	25	29	5	9
Fuel	11	14	8	10
Miscellaneous	14	17	6	7

Calculate Fisher’s price index number and show that it satisfies both Time Reversal Test and Factor Reversal Test for data given below.

Commodities	Base Year		Current Year
	Price	Quantity	Price	Quantity
Rice	10	5	11	6
Wheat	12	6	13	4
Rent	14	8	15	7
Fuel	16	9	17	8
Transport	18	7	19	5
Miscellaneous	20	4	21	3

Construct Fisher’s price index number and prove that it satisfies both Time Reversal Test and Factor Reversal Test for data following data.

Commodities	Base Year		Current Year
	Price	Quantity	Price	Quantity
Rice	40	5	48	4
Wheat	45	2	42	3
Rent	90	4	95	6
Fuel	85	3	80	2
Transport	50	5	65	8
Miscellaneous	65	1	72	3

Calculate the cost of living index number for the following data.

Commodities	Quantity 2005	Price
		2005	2010
A	10	7	9
B	12	6	8
C	17	10	15
D	19	14	16
E	15	12	17

Calculate the cost of living index number for the year 2015 with respect to base year 2010 of the following data.
\(\begin{array}{|c|c|c|c|} \hline \text { Commodities } & \begin{array}{c} \text { Number of } \\ \text { Units (2010) } \end{array} & \begin{array}{c} \text { Price } \\ (2010) \end{array} & \begin{array}{c} \text { Price } \\ (2015) \end{array} \\ \hline \text { Rice } & 5 & 1500 & 1750 \\ \hline \text { Sugar } & 3.5 & 1100 & 1200 \\ \hline \text { Pulses } & 3 & 800 & 950 \\ \hline \text { Cloth } & 2 & 1200 & 1550 \\ \hline \text { Ghee } & 0.75 & 550 & 700 \\ \hline \text { Rent } & 12 & 2500 & 3000 \\ \hline \text { Fuel } & 8 & 750 & 600 \\ \hline \text { Misc } & 10 & 3200 & 3500 \\ \hline \end{array}\)

Calculate the cost of living index number by consumer price index number for the year 2016 with respect to base year 2011 of the following data
\(\begin{array}{|c|c|c|c|} \hline & {\text { Price }} & \\ \begin{array}{c} \text { Commodities } \\ \text { } \end{array} & \begin{array}{c} \text { Base } \\ \text { year } \end{array} & \begin{array}{c} \text { Current } \\ \text { year } \end{array} & \text { Quantity } \\ \hline \text { Rice } & 32 & 48 & 25 \\ \hline \text { Sugar } & 25 & 42 & 10 \\ \hline \text { Oil } & 54 & 85 & 6 \\ \hline \text { Coffee } & 250 & 460 & 1 \\ \hline \text { Tea } & 175 & 275 & 2 \\ \hline \end{array}\)

Construct the cost of living index number for 2011 on the basis of 2007 from the given data using family budget method.

Commodities	Price		Weights
	2007	2011
A	350	400	40
B	175	250	35
C	100	115	15
D	75	105	20
E	60	80	25

The data shows the sample mean and range for 10 samples for size 5 each. Find the control limits for mean chart and range chart.

Sample	1	2	3	4	5	6	7	8	9	10
Mean	21	26	23	18	19	15	14	20	16	10
Range	5	6	9	7	4	6	8	9	4	7

The following data gives readings of 10 samples of size 6 each in the production of a certain product. Draw control chart for mean and range with its control limits.

Sample	1	2	3	4	5	6	7	8	9	10
Mean	383	508	505	582	557	337	514	614	707	753
Range	95	128	100	91	68	65	148	28	37	80

You are given below the values of sample mean ( \(\bar{X}\) ) and the range ( R ) for ten samples of size 5 each. Draw mean chart and comment on the state of control of the process.

Sample number	1	2	3	4	5	6	7	8	9	10
\(\overset{-}{X}\)	43	49	37	44	45	37	51	46	43	47
R	5	6	5	7	7	4	8	6	4	6

Given the following control chart constraint for : n = 5, A₂ = 0.58, D₃ = 0 and D₄ = 2.115

Compute the average seasonal movement for the following series

Year	Quarterly Production
	I	II	III	IV
2002	3.5	3.8	3.7	3.5
2003	3.6	4.2	3.4	4.1
2004	3.4	3.9	3.7	4.2
2005	4.2	4.5	3.8	4.4
2006	3.9	4.4	4.2	4.6

The annual production of a commodity is given as follows :
\(\begin{array}{|c|c|} \hline \text { Year } & \text { Production (in tones) } \\ \hline 1995 & 155 \\ \hline 1996 & 162 \\ \hline 1997 & 171 \\ \hline 1998 & 182 \\ \hline 1999 & 158 \\ \hline 2000 & 180 \\ \hline 2001 & 178 \\ \hline \end{array}\)
Fit a straight line trend by the method of least squares.

Determine the equation of a straight line which best fits the following data

Year	2000	2001	2002	2003	2004
Sales(Rs.000)	35	36	79	80	40

Compute the trend values for all years from 2000 to 2004

The sales of a commodity in tones varied from January 2010 to December 2010 as follows:

In year 2010	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
Sales (in tones)	280	240	270	300	280	290	210	200	230	200	230	210

Fit a trend line by the method of semi-average.

Use the method of monthly averages to find the monthly indices for the following data of production of a commodity for the years 2002, 2003 and 2004.

2002	15	18	17	19	16	20	21	18	17	15	14	18
2003	20	18	16	13	12	15	22	16	18	20	17	15
2004	18	25	21	11	14	16	19	20	17	16	18	20

Calculate the seasonal indices from the following data using the average from the following data using the average method:

	I Quarter	II Quarter	III Quarter	IV Quarter
2008	72	68	62	76
2009	78	74	78	72
2010	74	70	72	76
2011	76	74	74	72
2012	72	72	76	68

The following table shows the number of salesmen working for a certain concern:

Year	1992	1993	1994	1995	1996
No. of salesmen	46	48	42	56	52

Use the method of least squares to fit a straight line and estimate the number of salesmen in 1997.

Calculate price index number for 2005 by
(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

Compute
(i) Laspeyre’s
(ii) Paasche’s
(iii) Fisher’s Index 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

Using the following data, construct Fisher’s Ideal index and show how 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

Using Fisher’s Ideal Formula, compute price index number for 1999 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

Calculate Fisher’s index number to the following data. Also show that it satisfies Time Reversal Test.

Commodity	Price in Rupees per unit		Number of units
	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

A machine is set to deliver packets of a given weight. Ten samples of size five each were recorded. Below are given relevant data:

Sample number	1	2	3	4	5	6	7	8	9	10
\(\overset {-}{X}\)	15	17	15	18	17	14	18	15	17	16
R	7	7	4	9	8	7	12	4	11	5

Calculate the control limits for mean chart and the range chart and then comment on the state of control. (conversion factors for n = 5, A₂ = 0.58, D₃ = 0 and D₄ = 2.115

Ten samples each of size five are drawn at regular intervals from a manufacturing process. The sample means ( \(\overset{-}{X}\) ) and their ranges (R ) are given below:

Sample number	1	2	3	4	5	6	7	8	9	10
\(\overset {-}{X}\)	49	45	48	53	39	47	46	39	51	45
R	7	5	7	9	5	8	8	6	7	6

Calculate the control limits in respect of \(\overset {-}{X}\) chart. (Given A₂ = 0.58, D₃ = and D₄ = 2.115)  Comment on the state of control.

Construc \(\overset {-}{X}\) and R charts for the following data:

Sample Number	Observations
1	32	36	42
2	28	32	40
3	39	52	28
4	50	42	31
5	42	45	34
6	50	29	21
7	44	52	35
8	22	35	44

( Given for n = 3, A₂ = 0.58,D₃ = 0 and D₄ = 2.115)

The following data show the values of sample mean (\(\overset{-}{X}\)) and its range (R) for the samples of size five each. Calculate the values for control limits for mean, range chart and determine whether the process is in control.

Sample number	1	2	3	4	5	6	7	8	9	10
Mean	11.2	11.8	10.8	11.6	11.0	9.6	10.4	9.6	10.6	10.0
Range	7	4	8	5	7	4	8	4	7	9

( conversion factors for n = 5, A₂ = 0.58, D₃ = 0 and D₄ = 2.115)

A quality control inspector has taken ten samples of size four packets each from a potato chips company. The contents of the sample are given below, Calculate the control limits for mean and range chart.

Sample Number	Observations
	1	2	3	4
1	12.5	12.3	12.6	12.7
2	12.8	12.4	12.4	12.8
3	12.1	12.6	12.5	12.4
4	12.2	12.6	12.5	12.3
5	12.4	12.5	12.5	12.5
6	12.3	12.4	12.6	12.6
7	12.6	12.7	12.5	12.8
8	12.4	12.3	12.6	12.5
9	12.6	12.5	12.3	12.6
10	12.1	12.7	12.5	12.8

(Given for n = 5,  A₂ = 0.58, D₃  = 0 and D₄ = 2.115)

The following data show the values of sample means and the ranges for ten samples of size 4 each. Construct the control chart for mean and range chart and determine whether the process is in control

Sample number	1	2	3	4	5	6	7	8	9	10
\(\overset{-}{X}\)	29	26	37	34	14	45	39	20	34	23
R	39	10	39	17	12	20	05	21	23	15

In a production process, eight samples of size 4 are collected and their means and ranges are given below. Construct mean chart and range chart with control limits.

Sample number	1	2	3	4	5	6	7	8
\(\overset{-}{X}\)	12	13	11	12	14	13	16	15
R	2	5	4	2	3	2	4	3

In a certain bottling industry the quality control inspector recorded the weight of each of the 5 bottles selected at random during each hour of four hours in the morning.

Time	Weights in ml
8:00 AM	43	41	42	43	41
9:00 AM	40	39	40	39	44
10:00 AM	42	42	43	38	40
11:00 AM	39	43	40	39	42

Fit a straight line trend by the method of least squares to the following data.

Year	1980	1981	1982	1983	1984	1985	1986	1987
Sales	50.3	52.7	49.3	57.3	56.8	60.7	62.1	58.7

Calculate the Laspeyre’s, Paasche’s and Fisher’s price index number for the following data. Interpret on the data.

Commodities	Base Year		Current Year
	Price	Quantity	Price	Quantity
A	170	562	72	632
B	192	535	70	756
C	195	639	95	926
D	187	128	92	255
E	185	542	92	632
F	150	217	180	314
7	12.6	12.7	12.5	12.8
8	12.4	12.3	12.6	12.5
9	12.6	12.5	12.3	12.6
10	12.1	12.7	12.5	12.8

Using the following data, construct Fisher’s Ideal Index Number and Show that it satisfies Factor Reversal Test and Time Reversal Test?

Commodities	Price		Quantity
	Base Year	Current year	Base Year	Current year
Wheat	6	10	50	56
Ghee	2	2	100	120
Firewood	4	6	60	60
Sugar	10	12	30	24
Cloth	8	12	40	36

From the following data, calculate the control limits for the mean and range chart.

Sample No.	1	2	3	4	5	6	7	8	9	10
Sample Observations	50	51	50	48	46	55	45	50	47	56
	55	50	53	53	50	51	48	56	53	53
	52	53	48	50	44	56	53	54	49	55
	49	50	52	51	48	47	48	53	52	54
	54	46	47	53	47	51	51	57	54	52

The following data gives the average life(in hours) and range of 12 samples of 5 lamps each. The data are

Sample No	1	2	3	4	5	6
Sample Mean	1080	1390	1460	1380	1230	1370
Sample Range	410	670	180	320	690	450
Sample No	7	8	9	10	11	12
Sample Mean	1310	1630	1580	1510	1270	1200
Sample Range	380	350	270	660	440	310

Construct control charts for mean and range. Comment on the control limits.

The following are the sample means and ranges for 10 samples, each of size 5. Calculate the control limits for the mean chart and range chart and state whether the process is in control or not.

Sample number	1	2	3	4	5	6	7	8	9	10
Mean	5.10	4.98	5.02	4.96	4.96	5.04	4.94	4.92	4.92	4.98
Range	0.3	0.4	0.2	0.4	0.1	0.1	0.8	0.5	0.3	0.5