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.

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

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

The failure density function of a random variable T is given by


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

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). 

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:

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.

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. 

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:

 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.

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
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.

By generating 10 uniform random variate  estimate the integral

Recognizing this function as probability density function of , compare the value of  with .

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:

 Analyse the data and give your conclusions.

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

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

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:

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

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:

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.

State whether the following statements are true or false and also give the reason in support of your answer.
c) In SRSWOR, the possible numbers of sample of size n from a population of size N if sampling is done with replacement is Nⁿ
.d) One-way analysis of variance is a generalization of the two sample t-test.
e) If experimental error is reduced considerably and the efficiency of the design is
decreased.
f) If strata are heterogeneous then stratified sampling schemes provides estimates with greater precision.
g) If one wants to convert random numbers selected from two digit numbers 00-99 to
uniformly distributed U (0, 1) variables then one has to divide them by 99.

Which non-parametric test is used for testing the null hypothesis if it is known that the scores of the employees before and after the training programme do not follow the normal distribution but the distribution of the differences of scores before and after the training is symmetrical about its median? Conduct the test at 1% level of significance and conclude the result.

A company is trying to improve the work efficiency of its employees. It has organized a special training programme for all employees. In order to assess the effectiveness of the training programme, the company has selected 10 employees randomly and administered a well-structured questionnaire. The scores (out of 100) obtained by the employees are given in the following table:

To examine whether the training programme has improved efficiency of the employees,
give the answer of the following:
i) Are both samples are paired or independent?
ii) Formulate the null and alternative hypotheses.
iii) Which parametric test is used for testing the null hypothesis if it is known that the
scores of the employees before and after the training programme follow the normal
distribution? Conduct the test at 1% level of significance and conclude the result. 

. Complete the following table:

In a locality, 100 persons were randomly selected and asked about their educational achievements. The results were as follows:

Can we say that education depends on sex at 5% level of significance?

In a city, 36 out of a random sample of 500 men were found to drinkers at a certain date. After the heavy increase in tax on intoxicants, another sample of 100 men in the same city included 6 drinkers. Do you feel that the observed proportion of drinkers decreasing significantly at 1%level?