Define a semigroup. Give an example of an infinite semigroup.

Define a subgroup of a group. Check whether

is a subgroup of the the group of 2 × 3 matrices over ℂ under addition.

Define an abelian group. Give an example of a non-abelian group. (You don't need to prove that your example is a group. You have to only prove that it is non-abelian.)

Evaluate the integrausing beta and gamma functions.

Evaluate the integral

by considering D as a region of Type I and then as a region of Type II.

. be a function defined by Show that f is Riemann integrable using both definitions. Also, verify that the results of both definition match.

Find the equation of a line passing through points A(0, 0, 1) and B(1, 1, 0). Also, find the coordinates of a point on this line which is at a distance of 10 units from point A opposite to the side of point B.

In R we have a built-in data set “trees”. A screenshot of the last four rows together with the R code to obtain it is given as follows. To get more detail about this data set you can run ?trees command on R console.

Note that all the three variables of this data set are numeric. So, assuming each row of this data set is a point in 3-dimension. Find the distances between the points corresponding to the 28th and the 32st rows using the Manhattan and Chebyshev distance formula

Solve the following problems

(a) If f is a function from set X to a set Y then is it possible

In a class of Statistics, total number of students is 30. Select the linear and circular systematic random samples of 12 students. The age of 30 students is given below:

Age:     22     25        22      21       22        25      24      23     22      21      20     21      22 23     25     23     24     22    24   24     21     20

23       21     22       20      20     21      22    25

he 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:

Analyse the design with appropriate method and calculate the Critical Difference for the treatment mean yield

For the given data the yield of the treatment 2 in 3rd block is missing. Estimate the missing value and analyse the data.

A manurial trial with six levels of Farmyard 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 there is significant differences between the average mileage of 3 kinds of gasoline 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  is equal to

A population consists of 10 villages with a total of 212 households. The second column of the accompanying table shows the number of households corresponding to each village. Select a PPS with replacement sample of 6 villages by using the Cumulative Total method:

A sample of 100 employees is to be drawn from a population of collages A and B. The population means and population mean squares of their monthly wages are given below:

Village	Ni		2
Collage A	400	60	20
Collage B	200	120	80

Draw the 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 the that of SRSWOR.

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

(a) 

(b) The total number of all possible samples of size 3 without replacement from a population of size 7 is 21.

(c) While analysing the data of a 5 × 5 Latin Square design the d.f. for ESS is equal to 16

(d) In a Two-way Analysis of Variance test with 5 observations per cell having 4 blocks and 4 treatments the degree of freedom for the total variation is 64.

(e) The probability of selection of a sample of n from the population by SRSWOR is 1/ N

Consider the time series model

(i) Is this a stationary time series?

(ii) What are the mean and variance of the time series?

(iii) Calculate the autocorrelation function.

(iv)Plot the correlogram.

At a call centre, callers have to wait till an operator is ready to take their call. To monitor this process, 5 calls were recorded every hour for the 8-hour working day. The data below shows the waiting time in seconds:

(i) Use the data to construct control charts for mean and comments about the process. If process is out of control, then calculate the revised control limits.

(ii) If the specification limits as the 8±2, then calculate the process capability index Cpk and impetrate the result