Generate ten uniform random numbers U(0, 1) from the multiplicative LCG given below: x_i (5 x_i-1 ) mod 32 with x₀ = 1

Also obtain a sequence of heads and tails using above generated random numbers.

Times between successive crashes of a computer system were generated for a 6-months 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 value of parameter α and Mean (1/α) is given as 0.00435 and 230 hrs, respectively. Use kolmogrov-Smirnov test to examine the goodness of fit of exponential distribution.

For a factorial experiment with 3 factors, N, P and M each at two levels, the design and yield per plot are given below. Analyse the experiment.

Replicate – I                            

Replicate – II

Replicate - III

The kinetic energy of an electron is 13.6 eV. Calculate its de Broglie wavelength.

Estimate the minimum kinetic energy a neutron confined to a nucleus of diameter 4 10^-15 m may have.

The following data is the data pertaining to a feeding trial on sheep. Treatments

A: Grazing only

B: Grazing + Maize Supplements

C: Grazing + Maize + Protein Supplement P₁

D: Grazing + Maize + Protein Supplement P₂

E: Grazing + Maize + Protein Supplement P₃

Layout and Wool Yield (100 gm) is given as:

i) Analyse the design with appropriate method and calculate the CD for the treatment mean yield.

ii) Calculate the relative efficiencies of the above L.S.D over R.B.D and C.R.D.

The kinetic energy of an electron is equal to its rest mass energy. Determine the magnitude of its velocity and momentum.

A light source emits light of wavelength 125 nm while at rest. When the source is moving, the Doppler shifted wavelength of the emitted light is 375 nm. Is the source of light approaching us or receding from us? Also calculate its speed.

In the following data two values are missing. Estimate these values and analyse the design given as:

A man on the Moon observes two spaceships moving towards him from opposite directions at speeds of 0.7c and 0.8c, respectively. What is the relative speed of the two space ships as measured by an observer on either one?

A space ship is measured to be 140 m long on the Earth. When in flight, its length is measured as 120 m by an observer on the Earth. What is its speed?

A muon produced in the Earth’s atmosphere is travelling with a speed of 0.90c. As measured in the muon’s frame of reference, it has a lifetime of 1.6 s. What is its lifetime as measured by an observer on Earth?

A manurial trial with six levels of Farm Yard. Manure (FYM) was carried out in a randomised block design with 4 replications at the experimental station Junagarh with a new study the rate of decomposition of organic matters in soil and its synthetic capacity in soil on cotton crop. The yield per plot in kg for different levels of FYM and replications is given below:

                                                 Cotton Yield Per Plot (in Kg)

Carry out the Analysis of variance test and draw the conclusions

In order to compare the mileage yields of 3 kinds of gasoline, several tests were run and the following results were obtained:

Carry out the analysis of variance test and test whether the observed differences between the means of 3 kinds of gasoline may be attributed to chance at 5% level of significance.

In a population of size N = 5, the values of the population characteristics are 1, 3, 5, 7, 9, a sample of size 2 is drawn. Verify that is an unbiased estimate of and V() is equal to

From the following data find mean and variance of the stratified sampling under (i) proportional and (ii) Neyman Allocation

Strata	Ni	Ni	Si
1	80	29	12	80
2	160	39	8	30
3	260	32	4	10
	500	100

Explain by considering a sample of size n = 2 from a population consisting of five members 2, 3, 6, 8, 11 that SRSWOR gives a better estimator of population mean that SRSWR.

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

b) In cluster sampling the variance within clusters is greater than the variance between
clusters.
c) While analysing the data of a 5 × 5 Latin Square design the error d.f. is equal to 12.
d) In a Two way analysis of variance with 4 blocks & 4 treatments the degree of freedom
for the total variation is 15.
e) Suppose a random number generated by Middle. Square Method is 16, then the next
random number will be 44.

A chemist developing insect repellents wishes to know if a newly developed formula gives greater protection from insect bites than that given by the leading product on the market. In the experiment, 14 volunteers each had one arm spayed with the old product and the other sprayed with the new formula. Then each subject placed his arms into two chambers filled with equal number of mosquitoes, gnats and other biting insects. The numbers of bites received on each arm are as follows:

To test the new formula is more effective than the old one:

(i) State null and alternative hypotheses

(ii) Can you apply both parametric and non-parametric tests in this problem and why?

(iii)Write the assumptions to apply the suitable parametric test.

(iv)Apply the parametric test by assuming the assumptions write in part (iii) are fulfilled and write the conclusion.

(v) Apply the non-parametric test and write the conclusion.

(vi)Compare the conclusions drawn in parts (iv) and (v)

If magnitude of earthquakes recorded in a region of a country follows a distribution with parameter θ whose pdf is given below:    then

(i) Show that the estimators of the parameter θusing maximum likelihood and method of moments are same,

(ii) Show that maximum likelihood estimator is unbiased and sufficient forthe parameter θ.