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.

A company conducted a study on its employees to see the relationship of several variables with an employee’s IQ. For this purpose, fifteen employees were selected, and an IQ test and five different personality tests were given to them. Each employee’s IQ was recorded along with scores on five tests. The data are shown in the following table

Determine the most appropriate regression model for the employee’s IQ using a stepwise approach at a 5 % level of significance and interpret the results. Does the final regression model satisfy the linearity and normality assumptions?

For the data given in Question 2(b), obtain the followings:

(i) Diagonal of the hat matrix. Also, check the leverage points, if any.

(ii) Cook’s Distances. Also, verify the influence points, if any.

Explain the assumptions underlying the multiple linear regression model.

Suppose a researcher wants to evaluate the effect of cholesterol on the blood pressure. The following data on serum cholesterol (in mg/dL) and systolic blood pressure (in mm/Hg) were obtained for 15 patients to explore the relationship between cholesterol and blood pressure:

(i) Fit a linear regression model using the method of least squares.

(ii) Construct the normal probability plot for the regression model fitted on serum cholesterol and systolic blood pressure.

(iii) Test the significance of the fitted regression model.

Write a short note on the problem of multicollinearity and autocorrelation.

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

(i) We define three indicator variables for an explanatory variable with three categories.

(ii) If the coefficient of determination is 0.833, the number of observations and explanatory variables are 12 and 3, respectively, then the Adjusted R² will be 0.84.

(iii) For a simple regression model fitted on 15 observations, if we have h₁ = 0.37, then it is an indication to trace the leverage point in the regression model.

(iv) In a regression model  is not rejected, then the variable X₁ will remain in the model.

(v) The logit link function is log [-log(1-π)].