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Question:

Solve the following LPP using Simplex method:
Maximize $Z = x_1 + 2x_2$
Subject to the Constraints: $- x_1 + 2x_2 \leq 8$
$x_1 + 2 x_2 \leq 12$
$x_1 - x_2 \leq 3$
$x_1, x_2 \geq 0$

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Question:

Solve the following LPP using graphical method:
Maximize $Z = 3x_1 + 2x_2$
Subject to the Constraints:
$-2x_1 + x_2 \leq1$
$x_1 \leq 2$
$x_1 + x_2 \leq 3$
$x_1, x_2 \geq0$

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Question:

Rewrite the following linear programming problem in Standard form:
Minimise $Z = 2x_1 + x_2 + 4x_3$
Subject to the Constraints:
$-2x_1 + 4x_2 \leq 4$
$x_1 +2x_2 + x_3 \geq 5$
$2x_1 + 3x_3 \leq 2$
$x_1 \geq 0, x_2 \geq 0, x_3 \geq 0$

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Question:

State whether the following statements are True or False and also give the reason in support of your answer.
a) The Set $ ={(x, y) : 0 ≤ y ≤ 5 when 0 ≤ x≤ 2 and 3≤ y≤ 5 when 2≤ x ≤ 7 } is not a convex set.
b) If 10 is added to each of the entries of the cost matrix of a 3 x 3 assignment problem, then the total cost of an optimal assignment for the changed cost matrix will increase by 10.
c) The solution to a transportation problem with m-rows (supplies) and n-columns
(destinations) is feasible if number of positive allocations is m + n.
d) The Value d_i≥ 3 indicates an outlying observation in regression analysis.
e) Variations which occur due to natural forces and operate in a regular and periodic manner over a span of less than or equal to one year are termed as cyclic variations.

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Question:

The failure data of 10 electronic components are shown in the table given below:

Failure Number			1	2	3	4	5		6	7		8	9	10
Operating Time (in hours)			3	5	31	51	76		116	140		182	250	302

Estimate, the
i) reliability.
ii) cumulative failure distribution.
iii) failure density.
iv) failure rate functions.

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Question:

A small electronic device is designed to emit a timing signal of 200 milliseconds (ms) duration. In the production of this device, 10 subgroups of four units are taken at periodic intervals and tested. The results are shown in the following table:

Subgroup Number	Duration of Automatic Signal (in ms)
	a	b	c	d
1	195	201	194	201
2	204	190	199	195
3	195	197	205	201
4	211	198	193	180
5	204	193	197	200
6	200	202	195	200
7	196	198	197	196
8	201	197	206	207
9	200	202	204	192
10	203	201	209	192

i) Estimate the process mean and standard deviation.

ii) Determine the centre line and control limits for the process mean and process variability.

iii) By plotting the charts on graph paper, determine that the process is stable or not with respect to the process mean and process variability. If necessary, compute revised control limits

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Question:

The system shown below is made up of ten components. Components 3, 4 and 5 are not identical and at least one component of this group must be available for system success. Components 8, 9 and 10 are identical and for this particular group it is necessary that two out of the three components functions.

Image ignouassignments-ignouacademy-com--p-ignou-76901

What is the system reliability if R₁ = R₃ = R₅ = R₇ = R₉ = 0.85 and R₂ = R₄ = R₆= R₈ = R₁₀ = 0.95

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Question:

Solve the two-person zero-sum game having the following payoff matrix for player A:

		Player B
		B₁	B₂	B₃	B₄	B₅
Player A	A₁	3	4	5	–2	3
Player A	A₂	1	6	–3	3	7

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Question:

The failure density function of a random variable T is given by
$f(t) = \begin{Bmatrix} 0.011e^{-0.011t} &, t \geq 0 \\ 0 & , otherwise \end{Bmatrix}$

Calculate, the
i) reliability of the component.
ii) reliability of the component for a 100 hour mission time.
iii) mean time to failure (MTTF).
iv) median of the random variable T.
v) life of the component, if the reliability of 0.96 is desired

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Question:

A shirt manufacturing company supplies shirts in lots of size 250 to the buyer. A single
sampling plan with n = 20 and c = 1 is being used for the lot inspection. The company and the buyer decide that AQL = 0.04 and LTPD = 0.10. If there are 15 defective in each lot,
compute the
i) probability of accepting the lot.
ii) producer’s risk and consumer’s risk.
iii) average outgoing quality (AOQ), if the rejected lots are screened and all defective shirts are replaced by non-defectives.
iv) average total inspection (ATI).

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Question:

To monitor the manufacturing process of mobile phones, a quality controller randomly selected 100 mobile phones from the production line, each day over 15 days. The mobile phones were inspected for defectives and the number of defective mobile phones found each day was recorded. The data are given below:

Subgroup Number	Number of Mobile Phones Inspected	Number of Defective Mobile Phones
1	100	3
2	100	6
3	100	4
4	100	6
5	100	20
6	100	2
7	100	6
8	100	7
9	100	3
10	100	0
11	100	6
12	100	15
13	100	5
14	100	7
15	100	6

i) Determine the trial centre line and control limits for the fraction defective using the above data.
ii) Contract the control chart on graph paper and determine that the process is stable or not.
If there is any out-of-control point, determine the revised centre line and control limits.

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Question:

State whether the following statements are True or False. Give reason in support of your answer.
a) If the average number of defects in an item is 4, the upper control limit of the c-chart will be 12.
b) The specification limits and natural tolerance limits are same in statistical quality control.
c) If the probability of making a decision about acceptance or rejection of a lot on the first sample is 0.80 and the sizes of the first and second samples are 10 and 15, respectively, then the average sample number for the double sampling plan will be 25.
d) Two independent components of a system are connected in series configuration. If the reliabilities of these components are 0.1 and 0.30, respectively then the reliability of the system will be 0.65.
e) A point in the pictorial representation of a decision tree having states of nature as immediate sub-branches is known as decision point.

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Question:

A manufacturer wishes to determine the effectiveness of four types of machines (A, B, C and D) in the production of bolts. To accumulate this, the numbers of defective bolts produced for each of two shifts in the results are shown in the following table:

Machine	First shift					Second Shift
Machine	M	T	W	Th	F	M	T	W	Th	F
A	6	4	5	5	4	5	7	4	6	8
B	10	8	7	7	9	7	9	12	8	8
C	7	5	6	5	9	9	7	5	4	6
D	8	4	6	5	5	5	7	9	7	10

Perform an analysis of variance to determine at 5% level of significance, whether there is a difference (a) Between the machines and (b) Between the shifts.

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Question:

A sample of 100 villagers is to be drawn from a population of villages A and B. The population means and population mean squares of their monthly wages are given below:

Village	$N_i$	$\overline{X_i}$	$S_i^2$
Collage A	400	60	20
Collage B	200	120	80

Find the number of samples using Proportional and Neyman allocation techniques and compare. Obtain the sample mean and variances for the Proportional Allocation and SRSWOR for the given information. Then Find the percentage gain in precision of variances of sample mean under the proportional allocation over that of SRSWOR.

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Question:

By generating 10 uniform random variate $U (0, 1)$ estimate the integral

$\theta = \frac{1}{\sqrt{2\pi}}\int_{-1}^{2}e^{\frac{-x^2}{2}} dx$

Recognizing this function as probability density function of $N (0, 1)$ , compare the value of $\widehat{\theta}$ with $\theta$ .

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Question:

An experiment was planned to study the effect of Sulphate, Potash and Super Phosphate on the yield of potatoes. All the combinations of 2 levels of Super Phosphate [0 cent (p0) and 5 cent (p1)/ acre] and two levels of Sulphate and Potash [0 cent (k0) and 5 cent (k1)/acre] were studied in a randomised block design with 4 replications each. The (1/70) yields [lb per plot = (1/70) acre] obtained are given in table below:

Blocks

Yields (lbs per plot)

I

(1)
23

k

25

p

22

kp

38

II

p

40

(1)
26

k

36

kp

38

III

(1)
29

k

20

pk

30

p

20

IV

kp

34

k

31

p

24

(1)
28

Analyse the data and give your conclusions.

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Question:

Three varieties A, B and C of wheat are shown in five plots each of the following fields per acre as obtained:

Plots	A	B	C
1	8	7	12
2	10	5	9
3	7	10	13
4	14	9	12
5	11	9	14

Set up a table of analysis of variance and find out whether there is significant difference between the fields of these varieties.

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Question:

To determine the yield rate of wheat in a district of Punjab, 6 groups were constructed of 6 plots each. The data is given in the following table:

Plot No	Group 1	Group 2	Group 3	Group 4	Group 5	Group 6
1	8	6	18	13	17	12
2	13	5	8	7	15	15
3	11	16	6	13	10	11
4	26	5	10	6	21	17
5	13	16	16	7	20	8
6	31	5	20	2	25	10

Select a cluster sample of 3 clusters from the given data and find sample mean.

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Question:

In a class of Statistics, total number of students is 30. Select the linear and circulur systematic random samples of 10 students. The age of 30 students is given below:

Age:	22	25	22	21	22	25	24 23	22	21 20	21
22	23	25	23	24	22	24	24 21 20	23	21	22
20	20	21	22	25

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Question:

Assume that you have to perform a sample survey for Family expenditure of the faculty of Indira Gandhi National Open University. Then explain the main steps involved in the planning and execution of that sample survey.

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