🏛️ UPSC Advanced Examination Practice

📌 Q13 (e) Z-Test for Population Mean (10 Marks)

📊 Hypothesis Formulation

Test statistic for a single mean when variance is known:

💻 R Code

x <- c(168, 170, 185, 172, 169, 158, 170, 175)
sigma <- 5
mu0 <- 140
critical_value <- 1.96

n <- 0
sum_x <- 0

for(val in x) {
  n <- n + 1
  sum_x <- sum_x + val
}

xbar <- sum_x / n
Z <- (xbar - mu0) / (sigma / (n^0.5))

abs_Z <- ifelse(Z < 0, -Z, Z)

cat("Sample Mean =", xbar, "\n")
cat("Calculated Z-statistic =", Z, "\n")

if(abs_Z > critical_value) {
  cat("Decision: Reject H0 since |Z| > 1.96.\n")
} else {
  cat("Decision: Fail to reject H0.\n")
}

📌 Q14 (a) Control Statements in C (25 Marks)

Control statements determine the flow of execution, classified into Decision Making, Looping, and Jump Statements.

• Decision Making: if, if-else, switch
• Looping: for, while, do-while
• Jump Statements: break, continue, goto, return

📌 Q14 (b) C Program to Obtain ANOVA Table for RBD (25 Marks)

📊 ANOVA Model & Sum of Squares

💻 C Program

#include <stdio.h>

int main() {
    int t, r, i, j;
    float Y[10][10], G=0, SST=0, SSTreat=0, SSBlock=0;
    float T[10]={0}, B[10]={0};

    printf("Enter treatments and blocks: ");
    scanf("%d %d", &t, &r);

    for(i=0; i<t; i++){
        for(j=0; j<r; j++){
            scanf("%f", &Y[i][j]);
            G += Y[i][j];
            T[i] += Y[i][j];
            B[j] += Y[i][j];
            SST += Y[i][j]*Y[i][j];
        }
    }

    int N = t*r;
    SST -= (G*G)/N;

    for(i=0; i<t; i++) SSTreat += (T[i]*T[i])/r;
    SSTreat -= (G*G)/N;

    for(j=0; j<r; j++) SSBlock += (B[j]*B[j])/t;
    SSBlock -= (G*G)/N;

    float SSError = SST - SSTreat - SSBlock;

    printf("ANOVA Table\n");
    printf("Treatment SS = %.2f\n", SSTreat);
    printf("Block SS = %.2f\n", SSBlock);
    printf("Error SS = %.2f\n", SSError);

    return 0;
}

📌 Q14 (c) Compute Regression Equation of Y on X (25 Marks)

📊 Mathematical Formulation

💻 C Program

#include <stdio.h>

int main() {
    int n, i;
    float x[100], y[100];
    float sumx=0, sumy=0, sumxy=0, sumx2=0;
    float a, b;

    printf("Enter number of observations: ");
    scanf("%d", &n);

    for(i=0; i<n; i++) {
        printf("Enter X and Y: ");
        scanf("%f %f", &x[i], &y[i]);
        sumx += x[i];
        sumy += y[i];
        sumxy += x[i]*y[i];
        sumx2 += x[i]*x[i];
    }

    b = (n*sumxy - sumx*sumy) / (n*sumx2 - sumx*sumx);
    a = (sumy/n) - b*(sumx/n);

    printf("Regression Equation: Y = %.2f + %.2fX", a, b);
    return 0;
}

📌 Q14 (d) Reading Matrices in R and Performing Matrix Operations (25 Marks)

📖 Introduction

Matrices are fundamental data structures used in statistical computing and numerical analysis. A matrix is a two-dimensional rectangular array of elements arranged in rows and columns. In statistical applications, matrices are widely used for solving systems of equations, linear regression models, multivariate analysis, and scientific computing.

The programming language R provides efficient tools for matrix manipulation. However, for deeper understanding of matrix algebra, it is important to implement these operations manually rather than relying on built-in functions.

In this answer, we demonstrate how to read matrices in R and perform important matrix operations such as addition, multiplication, transpose, determinant, and inverse without using built-in functions.

📥 Reading a Matrix in R

A matrix can be created by specifying the number of rows and columns and then reading elements from the user.

n <- as.integer(readline("Enter number of rows: "))
m <- as.integer(readline("Enter number of columns: "))

A <- matrix(0, n, m)

for(i in 1:n) {
  for(j in 1:m) {
    A[i,j] <- as.numeric(readline(paste("Enter element", i, j, ": ")))
  }
}

print("Matrix A:")
print(A)

➕ Matrix Addition

Matrix addition is defined only when both matrices have the same dimensions. If $A$ and $B$ are matrices of order $n \times m$, their sum is given by:

C <- matrix(0, n, m)

for(i in 1:n) {
  for(j in 1:m) {
    C[i,j] <- A[i,j] + B[i,j]
  }
}

print("Matrix Addition Result")
print(C)

✖️ Matrix Multiplication

Matrix multiplication is defined when the number of columns of the first matrix equals the number of rows of the second matrix.

C <- matrix(0, n, p)

for(i in 1:n) {
  for(j in 1:p) {
    for(k in 1:m) {
      C[i,j] <- C[i,j] + A[i,k] * B[k,j]
    }
  }
}

print("Matrix Multiplication Result")
print(C)

🔄 Transpose of a Matrix

The transpose of a matrix is obtained by interchanging rows and columns. If $A$ is an $n \times m$ matrix, then its transpose $A^T$ is an $m \times n$ matrix.

T_mat <- matrix(0, m, n)

for(i in 1:n) {
  for(j in 1:m) {
    T_mat[j,i] <- A[i,j]
  }
}

print("Transpose of Matrix")
print(T_mat)

🧮 Determinant of a Matrix (Generalized Algorithm)

The determinant is defined only for square matrices and is a scalar value that provides important information about the matrix. A recursive method using cofactor expansion can be used to compute the determinant.

determinant <- function(A) {
  n <- nrow(A)

  if(n == 1) {
    return(A[1,1])
  }

  if(n == 2) {
    return(A[1,1]*A[2,2] - A[1,2]*A[2,1])
  }

  det <- 0

  for(k in 1:n) {
    submatrix <- A[-1, -k]
    det <- det + ((-1)^(1+k)) * A[1,k] * determinant(submatrix)
  }

  return(det)
}

🔄 Inverse of a Matrix (General Method)

The inverse of a matrix $A$ exists only if the determinant is non-zero.

Where $adj(A)$ is the adjoint matrix. The following algorithm computes the inverse using cofactors.

inverse_matrix <- function(A) {
  n <- nrow(A)
  detA <- determinant(A)

  if(detA == 0) {
    print("Inverse does not exist")
    return(NULL)
  }

  cof <- matrix(0, n, n)

  for(i in 1:n) {
    for(j in 1:n) {
      submatrix <- A[-i, -j]
      cof[i,j] <- ((-1)^(i+j)) * determinant(submatrix)
    }
  }

  # Manual transpose for the adjoint matrix
  adj <- matrix(0, n, n)
  for(i in 1:n) {
    for(j in 1:n) {
      adj[j,i] <- cof[i,j]
    }
  }

  inv <- adj / detA
  return(inv)
}

📚 Paper Summary & Key Focus Areas

🎯 Core Concepts Tested in This Paper

• Statistical Computing in C: Translating heavy formulas (ANOVA partitioning, simple linear regression, calculating standard deviation/variance) into iterative C loops. Memory management via arrays is crucial here.
• Manual Matrix Algebra (R): The examiner wants to see an understanding of linear algebra basics—specifically cofactor expansion for determinants and adjoint matrices for inversion—without relying on R's solve() or det().
• Hypothesis Testing: Properly formatting a statistical test (like the Z-test) with clear null and alternative hypotheses before writing the R code.

📌 Points to Remember

• Cumulative Variables: In sums like $\sum X^2$, always initialize your accumulator variables (e.g., sumx2 = 0) before starting your loop to prevent garbage value accumulation.

🎯 Try These: Practice Questions

📝 Practice Question 1 (C Programming)

Write a C program to compute the standard deviation for grouped frequency data.

📝 Practice Question 2 (R Programming)

Write an R script to compute the trace of an $n \times n$ matrix manually using a loop. Also, add logic to check if the matrix is symmetric.