Describe the following:

(i) t-distribution

(ii) General procedure of Testing of hypothesis 

6. Let  be a random sample from the Exponential distribution with pdf

(i) Obtain the maximum likelihood estimator (MLE) of .

(ii) Hence, find the maximum likelihood estimate of  for the observed sample: 

2.1, 1.6, 3.4, 0.9, 2.5

5. A researcher wants to compare the average test scores of students taught using Method A and Method B. A random sample of students was selected from each group, and their scores are given below:

Assuming that the populations are normally distributed with equal variances. For testing whether there is a significant difference between the mean scores of the two teaching methods at the 5% level of significance

(i) State the null and alternative hypotheses.

(ii) Name the appropriate and justify your choice.

(iii) Compute the test statistic and critical value.

(iv) State your conclusion. 

A survey was conducted to study the preference for online learning among students from urban and rural areas. Out of 300 urban students, 210 preferred online learning, while out of 250 rural students, 150 preferred online learning. Construct a 95% confidence interval for the difference between the population proportions of students preferring online learning in urban and rural areas. 
 

3. Let  be a random sample from a population with mean  and variance . Consider the estimator

(i) Find the bias of the estimator .

(ii) Find the variance of .
(iii) Check whether it is more efficient than sample mean. 

A school caretaker looks after 6 children. The ages (in years) of the children are given below:

(i) What is the nature (type) of the population of the ages of children?
(ii) Construct the sampling distribution of the sample mean for all possible samples of size .
(iii) Comment on whether the sampling distribution is approximately normal.
(iv) Find the mean and standard error of the sampling distribution. 

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

(a) The Wilcoxon rank-sum test is a non-parametric alternative to the two-sample t-test when normality is not assumed.

(b) The power of a statistical test is the probability of accepting the null hypothesis when it is true.

(c) If the sample size tends to infinity, the variance of a consistent estimator must tend to zero.

(d) In the method of moments, population moments are equated to corresponding sample moments.

(e) The chi-square distribution is symmetric for all degrees of freedom.

Times between successive crashes of a computer system were generated for a 6-month period and are given in increasing order as follows (time in hours):
1 10 20 30 40 52 63 70 80 90 100 102
130 140 190 210 266 310 530 590 640 1340
The parameter , mean .
Use the Kolmogorov-Smirnov test to examine the goodness of fit of exponential distribution. 

Generate a complete cycle for the LCG given below: , with . A man tosses an unbiased coin ten times. Using the first ten random numbers generated above, obtain a sequence of heads and tails by taking Head (H) as . 

The distribution function of Pareto distribution is given by , .
Given a , generate a random number from the above distribution, when  and .
Suppose , then find x. 

Identify the design given in the following table and then carry out the analysis:

In the following data, two values are missing. Estimate these values by Yates method and analyse the data by suitable technique.

A researcher wants to test four diets A, B, C, D on growth rate in mice. These animals are divided into 3 groups according to their weights. Heaviest 4, next 4 and lightest 4 are put in Block I, Block II, and Block III, respectively. Within each block, one of the diets is given at random to the animals. After 15 days, increase in weight is noted, which is given in the following table:

Perform a two-way ANOVA to test whether the data indicates any significant difference between' the four diets due to different blocks. 

The following data relate to production in kg of three varieties P, Q, R of wheat:

Is there any significant difference among the three varieties at 5% level of significance? 

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

Select a cluster sample of 3 clusters from the above data and find its sample mean. Further, explain the procedure of two-stage sampling if we want to draw a sample of 6 plots. Which are the 6 plots in your sample?

A sample of 60 students is to be drawn from a population consisting of 600 students belonging to two villages, A and B. The means and standard deviations of their marks are give below:

What are the sample sizes for the two villages using proportional allocation technique? 

 Draw all possible samples of size 2 from the population [2, 3, 4] and verify that . Also find variance of . 

State whether the following statements are true or false and also give the reason in support of your answer: 
(a) The total number of all possible samples of size 2 without replacement from a population of size 7 is 21.
(b) Consecutive 3 random numbers starting from 8937 by 'middle square method' are 8937, 8699, 6726.
(c) RBD is suitable in situations where it is not possible to divide the experimental material into a number of homogeneous blocks.
(d) As we increase the sample size, representativeness of the population by the sample decreases.
(e) In a big hall, there are 50 rows and each row has 60 students. A research scholar selects 10 rows randomly and then randomly selects 15 students from each selected row. It is an example of cluster sampling procedure.

A restaurant produces fresh burgers for its customers every day. The company is known for supplying fresh burgers and never uses burgers prepared on the previous day. Demand for burgers is uncertain, preparation capacity is limited, and the restaurant has the option of producing 0, 100, 200, 300 and 400 burgers every day. It has been estimated that the cost of producing each burgers pack is Rs.15. Each burger is sold for Rs. 20. Prepare a payoff matrix when 0, 100, 200, 300 or 400 demands of the burgers turn up on any given day. Prepare an opportunity loss table and hence find the optimum strategy.

In a manufacturing process, 4 items are inspected every hour for 10 consecutive hours. The measured quality characteristic (in mm) is given below:

Construct control chart for variability and mean and comment on the state of statistical control. If the process is out of control, obtain revised control limits.