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

a) In SRWOR, the possible numbers of sample of size \(\mathrm{n}\) from a population of size \(\mathrm{N}\) if sampling is done with replacement is \(\mathrm{N}^{\mathrm{n}}\).

b) One-way analysis of variance is a generalization of the two sample t-test.

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) 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.

2. 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.

3 a) 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:

\(\begin{array}{rrrrrrrrrrrrr}\text { Age: } & 22 & 25 & 22 & 21 & 22 & 25 & 24 & 23 & 22 & 21 & 20 & 21 \\ 22 & 23 & 25 & 23 & 24 & 22 & 24 & 2421 & 20 & 23 & 21 & 22 \\ 20 & 20 & 21 & 22 & 25 & & & & & & & & \end{array}\)

b) 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.

4. 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.

5. 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 \(\left[0\right.\) cent \(\left(\mathrm{p}_{0}\right)\) and 5 cent \(\left(\mathfrak{p}_{1}\right) /\) acre] and two levels of Sulphate and Potash [0 cent \(\left(\mathrm{k}_{0}\right)\) and 5 cent \(\left.\left(\mathrm{k}_{1}\right) / \mathrm{acre}\right]\) 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.

6. By generating 10 uniform random variate \(\mathrm{U}(0,1)\) estimate the integral

\[
\theta=\frac{1}{\sqrt{2 \pi}} \int_{-1}^{2} e^{-x^{2} / 2} d x
\]

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

7. A sample of 100 villagers is to be drawn from a population of villages A and \(\mathrm{B}\). The population means and population mean squares of their monthly wages are given below:

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. 8. A manufacturer wishes to determine the effectiveness of four types of machines \((\mathrm{A}, \mathrm{B}, \mathrm{C}\) 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.