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E commerce

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Types of cyberspace Architecture

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Question:

A division of climatic change is interested in analysing the pattern of the CO2 concentrations in the air of a particular state in past years and then forecasting the CO2 concentrations for the upcoming years. The monthly mean CO2 concentrations ppm (parts per million) mixing ratio in dry air from January 2006 through December 2019 were recorded. The following monthly data are given in the following table for past 14 years:

Month	2006	2007	2008	2009	2010	2011	2012	2013	2014	2015	2016	2017	2018	2019
January	299.16	299.62	300.60	301.67	303.71	305.00	306.77	308.32	309.90	310.41	312.87	313.88	315.04	316.33
February	299.94	300.40	301.60	302.17	304.23	305.63	307.26	309.41	310.70	311.68	313.59	314.62	315.70	316.82
March	300.41	300.87	302.57	303.86	305.54	306.93	309.10	310.69	311.70	312.04	314.11	316.23	316.38	318.29
April	301.72	302.18	303.72	305.07	306.79	307.95	309.88	311.51	312.65	314.27	316.07	316.62	318.38	319.91
May	302.13	302.92	303.68	305.82	306.95	308.05	310.47	312.02	313.28	314.92	316.38	317.53	318.93	319.41
June	301.09	302.43	303.17	305.12	307.00	308.27	310.31	311.54	312.42	314.40	315.78	316.87	318.26	318.74
July	300.10	300.85	301.96	303.81	305.37	306.64	308.41	309.88	311.02	313.16	313.96	315.00	316.44	316.92
August	298.14	299.01	299.80	301.56	303.47	304.68	306.74	307.75	308.97	311.11	311.71	312.86	314.55	315.03
September	296.36	297.51	297.98	300.30	301.46	302.77	305.07	306.05	306.84	309.11	309.86	311.55	313.21	313.69
October	296.29	297.41	297.57	300.22	301.29	303.09	305.07	306.13	307.00	309.15	310.13	311.57	312.67	313.15
November	297.23	298.25	299.20	301.37	302.76	304.29	306.22	307.45	308.20	310.38	311.84	313.11	314.09	314.57
December	298.55	299.97	300.58	302.49	303.80	305.76	307.38	308.85	309.63	312.02	313.32	314.49	315.27	315.75

a) Compute the seasonal indices using ratio to moving average method.
b) Obtain the deseasonalised values and then fit a linear trend line to the average annual CO2 concentrations using least squares method.
c) Convert the annual least-squared trend equation to a monthly trend equation.
d) Use the monthly trend equation and seasonal indices to forecast the CO2 concentrations for all twelve months of 2021.
e) Plot the original data, deseasonalised data and trend values.

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Question:

A Mobility-as-a-Service (MaaS) provider company conducted a study to check the relationship of several variables with its weekly commuters. For this purpose, thirty cities were selected and the number of weekly commuters were recorded along with other variables like: the average petrol price (in ₹), population of the city, monthly income of commuters (in ₹), average parking rates per month (in ₹). The data are given in the following table:

City	Number of Weekly Commuters	Average Petrol Price	Population of City (in ’000)	Average Monthly Income of Commuters (in ’00)	Average Monthly Parking Rates (in ₹)
1	17700	75	1900	580	1000
2	17540	75	1890	620	1000
3	17620	75	1880	640	1200
4	16260	77	1878	650	1200
5	16180	77	1850	655	1200
6	16340	77	1840	658	1400
7	16580	77	1825	820	1500
8	16020	82	1825	860	1500
9	15940	82	1820	880	1500
10	15892	82	1805	920	1600
11	15780	85	1810	963	1600
12	14820	95	1800	1057	1600
13	14660	95	1795	1133	1700
14	14660	96	1795	1160	2000
15	14580	96	1790	1180	2100
16	14420	96	1730	1183	2100
17	13380	97	1740	1265	2100
18	10070	120	1735	1300	2200
19	13220	102	1730	1325	2500
20	13540	102	1720	1380	2600
21	13700	102	1715	1401	3000
22	12100	107	1705	1450	3100
23	11124	113	1690	1500	3300
24	10900	125	1695	1520	3500
25	11108	114	1690	1560	3500
26	13668	104	1700	1600	3800
27	13780	90	1710	1620	4000
28	12108	118	1790	1590	3700
29	14668	108	1800	1630	4000
30	14780	94	1810	1650	4200

Now determine the most appropriate regression model for the number of weekly commuters using stepwise approach at 5 % level of significance and interpret the results. Does the final regression model satisfy the linearity and normality assumptions?

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Question:

An automobile manufacturing company examined the cars of a particular model to identify the number of defects during the final inspection stage. The total number of inspected carswere recorded for last 35 days along with the number of defects. The results are given in the following table:

Days	Total Inspected Cars	Number of Defects
1	45	6
2	35	2
3	40	1
4	30	2
5	40	5
6	25	1
7	35	3
8	30	4
9	30	6
10	40	2
11	35	11
12	35	1
13	25	3
14	40	7
15	45	2
16	40	6
17	30	5
18	35	2
19	30	7
20	45	3
21	28	1
22	38	4
23	33	11
24	33	1
25	43	4
26	38	2
27	38	5
28	28	4
29	43	6
30	48	1
31	43	3
32	33	4
33	38	3
34	33	2
35	48	5

Construct a suitable control chart for the number of defects to check whether the process is under statistical control or not. Also plot the revised control charts, if necessary.

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Question:

A Company wants to maintain the quality of bottling process which uses a particular brand of machine to fill 100 ml sanitizer spray bottle. During each shift, a sample of 10 bottles is selected (2 hours apart) and the volume of the each filled sanitizer spray bottle (in ml) is determined. In this regards, total 25 subgroups consisting of a sample of 10 bottles in each subgroup were selected. The following table lists the measurements from 25 consecutive shifts:

Sample No.	Obs. 1	Obs. 2	Obs. 3	Obs. 4	Obs. 5	Obs. 6	Obs. 7	Obs. 8	Obs. 9	Obs. 10
1	99.46	100.12	99.73	99.56	99.46	99.73	99.46	100.12	99.73	99.56
2	100.95	100.00	99.66	100.06	100.95	99.66	100.95	100.00	99.66	100.06
3	99.84	99.43	99.62	99.73	99.84	99.62	99.84	99.43	99.62	99.93
4	99.85	99.26	99.77	99.56	99.85	99.77	99.85	99.86	99.77	99.76
5	99.66	100.12	99.91	100.02	99.66	99.91	99.66	100.12	99.91	100.02
6	99.82	100.06	99.87	100.18	99.82	99.87	99.82	100.06	99.87	100.18
7	99.86	99.66	99.46	99.52	99.86	99.46	99.86	99.66	99.86	99.72
8	99.87	99.62	100.12	99.62	99.87	100.12	99.87	99.62	100.12	99.62
9	99.85	99.67	100.13	100.07	99.85	100.13	99.85	99.67	100.13	100.07
10	99.72	99.52	99.85	99.71	99.72	99.85	99.72	99.52	99.85	99.71
11	99.88	100.00	100.26	99.87	99.88	100.26	99.88	100.00	99.86	99.87
12	99.65	100.06	100.15	99.93	99.65	100.15	99.65	100.06	100.15	99.93
13	99.46	99.70	99.43	99.87	99.86	99.63	99.66	99.70	99.83	99.87
14	99.91	99.60	99.75	100.22	99.91	99.75	99.91	99.60	99.75	100.22
15	100.04	100.06	99.82	99.85	100.04	99.82	100.04	100.06	99.82	99.85
16	100.15	100.10	99.93	99.62	100.15	99.93	100.15	100.10	99.93	99.62
17	100.13	99.56	98.81	99.25	100.13	98.81	100.13	99.56	98.81	99.25
18	99.82	100.05	100.08	99.75	99.82	100.08	99.82	100.05	100.08	99.75
19	99.90	100.10	99.91	100.15	99.90	99.91	99.90	100.10	99.91	100.15
20	99.50	99.76	99.95	99.56	99.50	99.95	99.50	99.76	99.95	99.56
21	99.88	100.12	100.00	100.25	99.88	100.00	99.88	99.92	100.00	100.05
22	99.87	99.67	99.51	99.71	99.87	99.51	99.87	99.67	99.51	99.71
23	100.31	99.91	99.60	99.96	100.31	99.60	100.31	99.91	99.60	99.96
24	99.66	99.46	99.91	99.72	99.66	99.91	99.66	99.46	99.91	99.72
25	99.90	100.02	99.70	100.07	99.90	99.70	99.90	100.02	99.70	100.07

Construct suitable control charts for variability as well as for average to infer whether the process of bottling is under statistical control or not. If it is out-of-control, also plot the revised control charts, if necessary.

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Question:

Suppose that a production house wants to evaluate the popularity of eight books on a particular subject. The customer service manager of the production house hires seven evaluators with varying experience in that subject to review the books. To reduce the effect of the variability from evaluator to evaluator, she uses a randomised block design, with evaluators serving as the blocks. The eight books are the groups of interest.

The seven evaluators assigned to each of the eight books in a random order. A rating scale from 0 (low) to 100 (high) is used. The following table summarises the results:

	Books
Evaluators	A	B	C	D	E	F	G	H
1	68	59	80	72	66	57	78	79
2	75	73	86	74	73	71	84	85
3	74	65	88	78	72	63	80	86
4	78	61	85	74	76	59	83	84
5	82	64	90	82	80	62	88	89
6	76	66	92	84	74	64	78	90
7	75	73	86	74	73	71	84	82

The effect of each evaluator of eight books is normally distributed with approximately equal variances.
i) Analyse the design at 5% level of significance.
ii) Is the average popularity of the eight books significantly different? If the difference between the averages popularity of the eight books is significant, do the pair-wise comparison between them.

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Question:

For the data given in Question 1, compare the water wastage of both plants to get the answers of the following questions:
i) Is there enough evidence that the average water wastage of Plant A is more than the average water wastage of Plant B at 5 % level of significance?
ii) Are the variances of the distributions of water wastage of Plants A and B equal at 5 % level of significance?

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Question:

The number of employees (in hundreds) and the revenues (in ₹ hundred crores) of 20 companies were recorded to access the relationship between the revenue generated and strength of the employees. The data are given in the following table:

S. No.	No. of Employees (in '00)	Revenue (in ₹ '00 crores)
1	165	335
2	550	425
3	330	345
4	550	415
5	275	325
6	330	360
7	385	360
8	330	365
9	440	410
10	385	375
11	275	335
12	495	390
13	495	395
14	440	395
15	495	400
16	440	425
17	220	320
18	330	420
19	440	425
20	330	370

Compute the Spearman’s rank correlation coefficient between the number of employees and the revenues of the companies.

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Question:

The production of semiconductors needs a lot of water to cool down the equipment and clean silicon wafers. To study the water wastage during the production of semiconductor chips, the data of the water required for past 100 days in two plants: Plant A and Plant B are given in the following table:

Day	Plant A (in '000 litres)	Plant A (in '000 litres)	Day	Plant A (in '000 litres)	Plant A (in '000 litres)
1	616	670	51	910	1092
2	656	766	52	866	1040
3	780	910	53	672	806
4	728	850	54	704	846
5	814	650	55	810	972
6	648	756	56	854	1024
7	748	872	57	638	766
8	780	910	58	740	890
9	624	730	59	854	1024
10	642	750	60	650	780
11	764	892	61	684	712
12	814	950	62	632	760
13	656	766	63	752	902
14	678	790	64	702	842
15	858	1030	65	786	942
16	632	760	66	624	748
17	624	702	67	722	866
18	900	1080	68	752	902
19	684	728	69	642	722
20	726	872	70	664	742
21	912	1096	71	738	886
22	868	1042	72	786	942
23	674	810	73	632	760
24	708	850	74	652	784
25	812	976	75	852	1022
26	856	1028	76	626	752
27	640	768	77	692	696
28	744	892	78	894	1072
29	856	1028	79	652	722
30	652	782	80	722	866
31	774	716	81	908	1090
32	636	762	82	862	1036
33	754	906	83	668	802
34	704	846	84	702	842
35	788	946	85	808	970
36	626	752	86	850	1022
37	724	870	87	634	762
38	754	906	88	738	886
39	632	726	89	850	1022
40	622	746	90	646	776
41	740	890	91	704	710
42	788	946	92	630	756
43	636	762	93	750	900
44	656	788	94	700	840
45	854	1026	95	782	940
46	630	756	96	654	746
47	708	698	97	718	862
48	896	1076	98	750	900
49	624	726	99	628	720
50	724	870	100	702	740

Answer the followings:
i) Which plant has more wastage of water?
ii) Which plant shows greater variability in the wastage of water?
iii) Determine the correlation between the wastage in both plants.
iv) Compute suitable width of the class intervals for both plants.
v) Construct the continuous frequency distribution for both plants.

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Question:

Seven successive observations on a stationary time-series are as follows:
12, 14, 13, 10, 15, 12, 15
(a) Calculate auto-covariances C₀, C₁, C₂, C₃ and C₄.
(b) Calculate auto-correlation coefficients r₁, r₂, r₃ and r₄.
(c) Plot the correlogram.

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Question:

The following table represents the sales (in thousands) of mobile sets of a shop for 16 quarters over four years:

Year	Quarter
Year	^Q1	^Q2	^Q3	^Q4
2011	554	590	616	653
2012	472	501	521	552
2013	501	531	553	595
2014	403	448	460	480

(a) Compute the seasonal indices for four quarters by Simple average method.
(b) Obtain deseasonlised values.

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Question:

A researcher is interested in developing a linear model for the electricity consumption of a household having an AC (1.5 ton) so that she can predict the electricity consumption. For this purpose, she selects 25 houses and records the electricity consumption (in kWh), size of house (in square feet) and AC hours for one month during summers. The results obtained are:
b̂₀ = 22.381 b̂₁ = 1.6161, b̂₂ = 0.0144, SS( b̂0) = 12526.08, SS( b̂0, b̂1) = 17908.47, SS( b̂0, b̂2) = 17125.23, SS( b̂0, b̂1, b̂2) = 18079.0, σ²= 10.53, SE( b̂₁) = 0.17,= and SE( b̂₂) = 0.0035.

Build a regression model by selecting appropriate regressors in the model using the Stepwise Selection method.

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Question:

A firm wants to know whether there is any linear relationship between the sales (X) and its yearly revenue (Y). The records for 10 years were examined and the following results were obtained: ∑ X = 265, ∑Y = 27.73, ∑SS_x = 285.6, ∑ SS_Y = 6.978 and SS_XY = 57.456.
(a) Fit a regression line taking Y as the dependent variable and X as the independent variable.
(b) Test whether the sales have any effect on revenue at 5% level of significance.
(c) Comment on the goodness of fit of the regression line.

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Question:

Using the graphical method to minimise the time required to process Job 1 and Job 2 on five machines A, B, C, D and E, find the minimum elapsed times an idle times to complete both jobs.

Job 1	Sequence	A	B	C	D	E
	Time (in hours)	1	2	3	5	4
Job 2	Sequence	C	A	D	E	B
	Time (in hours)	3	4	2	1	5

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Question:

In a railway marshalling yard, goods trains arrive at a rate of 36 trains per day. Assuming that the inter-arrival and service time distributions both follow exponential distribution with an average of 30 minutes, calculate the following:
(i) Traffic intensity
(ii) The mean queue length
(iii) Probability that the queue size exceeds

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Question:

Four professors are capable of teaching any one of four different courses. Class preparation time in hours for different topics varies from professor to professor and is given in the table below:

Professor	A	B	C	D
Linear Programming	2	15	13	4
Queuing Theory	10	4	14	15
Transportation Problem	9	14	16	13
Regression Analysis	7	8	11	9

Each professor is assigned only one course. Determine an assignment schedule so as to minimise the total course preparation time for all courses.

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Question:

A company has three production facilities S₁ , S₂ and S₃ with production capacity of 7, 9 and 18 units (in 100s) per week of a product, respectively. These units are to be shipped to four warehouses D1, D2, D3 and D4 with requirement of 5, 6, 7 and 14 units (in 100s) per week, respectively. The transportation costs (in Rs) per unit between factories to warehouses are given in the table below:

	^Dl	^D2	^D3	^D4	Capacity
^S1	19	30	50	10	7
^S2	70	30	40	60	9
^S3	40	8	70	20	18
Demand	5	8	7	14	34

Obtain optimal solution by the MODI method.

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Question:

Use the penalty (Big M) method to solve the following LP problem:
Minimise Z = 5x₁+ 3x₂
Subject to the constraints
2x₁+ 4x₂ ≤ 12
2x₁+ 2x₂ = 10
5x₁+ 2x₂≥ 10
x₁, x₂≥ 0.

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Question:

State whether the following statements are True or False. Give reasons in support of your answers.
(a) The solution of a transportation problem with 5 rows (supplies) and 4 columns
(destinations) is feasible if number of possible allocations are 8.
(b) The moving averages of suitable period in a time-series are free from the influences of seasonal and cyclic variations.
(c) If the basic solutions for a system of equations are (- 2, 0, 1), (0, 1, 3), (- 2, 3, 0), then only (0, 1, 3) is feasible.
(d) In the stepwise selection method of multiple regression model, once a variable enters in the model then it always remains in the model.
(e) An enterprise requires 1000 units per month. The ordering cost is estimated to be 50 per order. The purchase price is 20 per unit and the carrying cost per unit is 10% of it. Then the economic lot size to be ordered is 775.

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