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

A system has seven independent components and reliability block diagram of it shown as follows:

Find reliability of the system.

A manufacturer of men's jeans purchases zippers in lots of 500. The jeans manufacturer uses single-sample acceptance sampling with a sample size of 10 to determine whether to accept the lot. The manufacturer uses c = 2 as the acceptance number. Suppose 3% nonconforming zippers are acceptable to the manufacturer and 8% nonconforming zippers are not acceptable. Find

(i) Probability of accepting a lot of incoming quality 0.04.

(ii) Average outing quality (AOQ), if the rejected lots are screened and all defective zippers are replaced by non-defectives.

(iii) Average total inspection (ATI).

Differentiate between the autoregressive and moving average models of time series.

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

(i) The R-chart is suitable when subgroup size is greater than 

(ii) In single sampling plan, if we increase acceptance number then the OC curve will be steeper.

(iii) If the effect of summer and winter is not constant on the sale of AC then we use the additive model of the time series.

(iv) If a researcher wants to find the relationship between today's unemployment and that of 5 years ago without considering what happens in between then the partial autocorrelation is the better way in comparison to autocorrelation.

(v) A system has four components connected in parallel configuration with reliability 0.2, 0.4, 0.5, 0.8. To improve the reliability of the system most, we have to replace the component which reliability is 0.2.

Create a data frame named RData consisting of the following data:

Write R commands to:

(i) Compute the group wise means of x, y and z according to the groups defined by w column using apply family function.

(ii) Sort RData according the y column of it.

Create following two matrices A and B with following elements.

Write R commands to do the following tasks:

(i) Multiply the two matrices.

(ii) Combine the two matrices row-wise.

(iii) Create a function that computes the following expression:

Create the graph of the following function.

Create an array of two dimension with following elements.

Also, extract the row shown in the rectangular box.

Create a date object named Ddata consisting of the following dates.

26Jan2023, 15Aug2023, 020ct2023, 05Sep2023

Create a function to compute ranks (in case of tied ranks) of the given data.

Write R command to create a data frame named DF with worker's data and do the following tasks:

(i) Use suitable function to remove NA from the data and then create a scatter plot.

(ii) Write the known data obtained in step (i) to a.txt file named "WORK".

The following data relates to the number of items produced per shift by two workers for a number  of days.

(a) Write R command to create a list named LT with worker's data. Also, after creating the list, do the following tasks:

(i) Use a suitable loop function to compute the mean of number of items produced by each worker in a single line command.

(ii) Extract the worker A data from it by using two different approaches.

Write the output of the following statements:

(i) rep(x=c(T, F, T, F), times=c(2,1,2,3))

(ii) 5%/%3; diag (3)

(b) Differentiate between the use of the sep and collapse arguments of the paste() function.

(c) Write R commands to create a bar plot of the following data by using arguments of the used function for filling up the bars and to give labels to the axis:

5, 10, 8, 7, 8, 5, 8, 7, 5, 8, 9, 6, 8, 8, 8

(d) Check whether the given loop is finite or infinite. If infinite, do the necessary changes in the written loop to make it finite.

x<-0

repeat{

print(x^2)

x<-x+1

if(x<5) print(x) }2.

The following data on the diagnosis of coronary heart disease (where 0 indicates absence and 1 indicates presence), serum cholesterol (in mg/dl), and weight (in kg) were obtained for 80 patients to explore the relationship of coronary heart disease with cholesterol and weight:

(i) Fit a multiple logistic model for the dependence of coronary heart disease on the  average serum cholesterol and weight considering  as the initial values of the parameters (solve only for one Iteration).

(ii) Test the significance of the fitted model using the Hosmer-Lemeshow test at 5% level of significance.