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: 

Based on the image provided, here is the data organized into a clear table:

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

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 49.

b) RBD is suitable in situations where it is not possible to divide the experimental material into a number of homogeneous blocks.

c) If experimental error is reduced considerably and the efficiency of the design is decreased.

d) If strata are heterogeneous then stratified sampling schemes provides estimates with greater precision.

e) As we increase the sample size, representativeness of the population by the sample decreases.

 Two independent groups of patients were given Drug X and Drug Y. The recovery times (in days) are as follows:

 (i) State the null and alternative hypotheses.

(ii) Explain why the Mann–Whitney U test is appropriate for this data.

(iii) Perform the test at the 5% level of significance and draw a conclusion. 

Suppose the number of breakdowns per month in a factory follows a Poisson distribution with parameter λ.

(i) Find the Cramer-Rao lower bound for an unbiased estimator of λ.

(ii) Obtain the UMVUE of λ.

(i) Find the Cramer-Rao lower bound for the variance. (ii) Also, find the UMVUE of $\lambda$. 

A pharmaceutical company fills medicine bottles with a target weight of 500 mg. It is known that the filling weights are normally distributed with known variance o² = 16. Derive the likelihood ratio test for testing

Ηο: μ = 500againstH₁: μ ≠ 500

at significance level a.

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

(i) When the population distribution is unknown and the data are measured on a nominal scale, the Sign test can be used to test a hypothesis about the median.

(ii) In the Bayesian framework, prior information about the parameter is combined with the likelihood to obtain the posterior distribution.

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

(iv) A complete sufficient statistic guarantees the uniqueness of the unbiased estimator obtained through it.

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

Descriptive and Experimental research.

Applied and Basic research

Quantitative and Qualitative research

What is a research report? Explain the types of research reports and discuss the principles of effective report writing.

Discuss research ethics. Explain issues related to plagiarism, informed consent, confidentiality, and ethical responsibility of researchers.

Discuss the methods of data collection. Explain primary and secondary data, their sources, advantages, and limitations.

What is a hypothesis? Explain the types of hypotheses. Describe the procedure for hypothesis testing in detail.

Explain Sampling design. Discuss the probability and non-probability sampling methods, highlighting their advantages, disadvantages, and suitability.

What is research design? Explain the various types of research designs-exploratory, descriptive, diagnostic, and experimental-with their merits and limitations.

Discuss in detail the research process. Explain each step involved in the research process with suitable examples.

 State whether the following statements are true or false and also give the reason in support of your answer.

(i) Research is a systematic and objective process of finding answers to scientific problems.

(ii) A hypothesis is always proved to be true at the end of a research study.

(iii) Exploratory research is conducted when little or no prior information is available about the problem.

(iv) Plagiarism refers to using someone else's work without proper acknowledgment.

(v) In qualitative research, numerical analysis plays a dominant role.

How will autonomy and financial independence help an IPO, considering has to function within the policy framework given by the national? Why cannot an IPO be completely free of government's control?

 Discuss the purpose, objectives and functioning of Patent Co-operation Treaty.