A system consists of six independent components arranged as follows:

• Two components with reliabilities 0.9 and 0.8 connected in series.

• This series combination is connected in parallel with a component of reliability 0.7.

• The resulting subsystem is connected in series with two components of reliabilities 0.85 and 0.95.

Draw the reliability block diagram and calculate the overall reliability of the system.

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

(i) Check whether saddle point exit or not.

(ii) If saddle point does not exit then determine optimal strategies for both the manufacturers and value of the game.

3. A company manufactures water pumps. The quality control inspector of the company takes a sample of 100 water pumps at regular intervals. The numbers of defective pumps for 15 samples are given below:

Use the data to construct a suitable chart. Observe the results and comment on the control of the process as indicated by the chart.

A factory purchases bolts in lots of 800. Acceptance is decided using a single-sampling plan with sample size n = 20 and acceptance number c = 3. Assume that 2% of defective items are considered acceptable quality and 7% defective items are considered unacceptable quality.

Find:

(i) The probability of accepting a lot when the incoming quality level is 5% defective.

(ii) The Average Outgoing Quality (AOQ), assuming rejected lots are completely screened and defectives are replaced.

(iii)The Average Total Inspection (ATI)

State whether the following statements are True or False. Give reason in support of your answer: 

(a) The c-chart is suitable for monitoring to proportion of defective.

(b) In single sampling plan, if we increase acceptance number then the OC curve will be steeper.

(c) In a series system, improving the reliability of the weakest component gives the maximum improvement in system reliability.

(d) If the probabilities are not associated with the occurrence of different states of nature, then the situation is known as decision making under risk.

(e) In single sampling plan, if we increase acceptance number then the OC curve will be steeper.

 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.

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

(a) Compute the seasonal indices for four quarters by Simple average method.

(b) Obtain deseasonlised values.

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

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:
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(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.

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.

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

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:

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

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:

Obtain optimal solution by the MODI method

 Use the penalty (Big M) method to solve the following LP problem:

Minimise 

Subject to the constraints



.

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.

Define experimental and non-experimental studies.

Explain the steps taken for minimising bias in a research setup of a clinical trial.

Differentiate between direct and indirect bioassays.

Explain any five types of biases in clinical trials.

(a) In a hypothetical study, 10 patients with kidney disease were randomly assigned to receive two dialysis methods (A and B). 5 patients received Dialysis-A followed by Dialysis-B, and the remaining 5 received Dialysis-B followed by Dialysis-A. The kidney function scores of these patients are given as:

Test whether the mean score for dialysis-A is different from the mean score of dialysis-B, assuming no carry-over effect.

(b) Describe descriptive and analytic epidemiology.