About this previous year paper
SECTION A – Short Answers (2 × 10 = 20 Marks)
(a) Significance of Measures of Dispersion
Measures of dispersion describe how data values are spread around the mean.
Help identify variability, reliability, and consistency of data.
Examples:
Range = Max − Min → simple but affected by outliers.
Standard Deviation (SD) → average distance from mean; best for normally distributed data.
Difference: SD is more robust; range gives only extremes.
(b) Concept of Mean
The mean (average) is the sum of all values divided by the number of observations:
Xˉ=∑Xin\bar{X} = \frac{\sum X_i}{n}Xˉ=n∑Xi
Advantages: Simple, widely used, uses all data points.
Limitations: Affected by extreme values (outliers).
(c) Correlation and Its Significance
Correlation measures the degree of relationship between two variables (X & Y).
Positive correlation: both increase together (e.g., height & weight).
Negative correlation: one increases while the other decreases (e.g., speed & travel time).
Significance: Helps in prediction and understanding relationships.
(d) Inference Procedure for Correlation Coefficient
Steps:
State H0:ρ=0H_0: ρ = 0H0:ρ=0 (no correlation). Compute sample correlation rrr.
Calculate test statistic: t=rn−21−r2t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}t=1−r2rn−2
Compare with t-critical value → accept/reject H0H_0H0.
Importance: Ensures correlation isn’t due to chance.
(e) Bivariate vs Simple Correlation
Simple correlation: between two variables (X, Y).
Bivariate correlation: includes simultaneous analysis of two or more related variables.
Example: Relationship between height–weight (simple) vs height–weight–age (bivariate).
(f) Linear Regression
Regression estimates the relationship between dependent (Y) and independent (X) variable:
Y=a+bXY = a + bXY=a+bX
where
b=n∑XY−(∑X)(∑Y)n∑X2−(∑X)2b = \frac{n\sum XY - (\sum X)(\sum Y)}{n\sum X^2 - (\sum X)^2}b=n∑X2−(∑X)2n∑XY−(∑X)(∑Y),
a=Yˉ−bXˉa = \bar{Y} - b\bar{X}a=Yˉ−bXˉ.
Slope (b) shows rate of change in Y per unit X.
(g) Simple vs Multiple Regression
| Feature | Simple | Multiple |
|---|---|---|
| Variables | 1 dependent, 1 independent | 1 dependent, ≥2 independent |
| Equation | Y = a + bX | Y = a + b₁X₁ + b₂X₂ + … |
| Use | Simple relations | Multivariate impact |
| Reason to use Multiple Regression: To study influence of several predictors. |
(h) Correlation Coefficient (X: 10,15,20,25; Y: 60,75,80,90)
r=nΣXY−(ΣX)(ΣY)[nΣX2−(ΣX)2][nΣY2−(ΣY)2]r = \frac{nΣXY - (ΣX)(ΣY)}{\sqrt{[nΣX^2 - (ΣX)^2][nΣY^2 - (ΣY)^2]}}r=[nΣX2−(ΣX)2][nΣY2−(ΣY)2]nΣXY−(ΣX)(ΣY)
After calculation:
r≈0.97r ≈ 0.97r≈0.97 → Strong positive correlation.
(i) Regression Line (Y on X) Given X = [2,4,6,8], Y = [5,8,11,14]:
Slope:
b=Σ(X−Xˉ)(Y−Yˉ)Σ(X−Xˉ)2=1.5b = \frac{Σ(X - \bar{X})(Y - \bar{Y})}{Σ(X - \bar{X})^2} = 1.5b=Σ(X−Xˉ)2Σ(X−Xˉ)(Y−Yˉ)=1.5
Intercept:
a=Yˉ−bXˉ=2a = \bar{Y} - b\bar{X} = 2a=Yˉ−bXˉ=2 Equation: Y=2+1.5XY = 2 + 1.5XY=2+1.5X
(j) Probability of Queen given Face Card Face cards = 12 (J,Q,K of 4 suits) → 4 queens.
P(Q∣F)=412=13P(Q|F) = \frac{4}{12} = \frac{1}{3}P(Q∣F)=124=31
SECTION B – Descriptive Questions (Any 3 × 10 = 30 Marks)
(a) Singular Value Decomposition (SVD)
Decomposes matrix BBB as B=UΣVTB = UΣV^TB=UΣVT
Σ (Sigma): diagonal matrix of singular values. U, V: orthogonal matrices.
Significance:
Helps in dimensionality reduction, noise removal, and data compression.
Retaining top-k singular values ≈ low-rank approximation.
(b) Multiple Regression Analysis
Model: Y=β0+β1X1+β2X2+εY = β_0 + β_1X_1 + β_2X_2 + εY=β0+β1X1+β2X2+ε
Interpretation: βiβ_iβi: effect of XiX_iXi on Y holding others constant.
Goodness of fit by R2R^2R2.
Example: predicting house price using area & location.
(c) Randomization Test (Two Teaching Methods)
Steps:
Combine all scores from both groups. Randomly reassign into new groups (15 & 20 students).
Calculate mean difference repeatedly (e.g., 10,000 trials).
Compare observed difference vs simulated distribution.
→ If p<0.05p < 0.05p<0.05, teaching methods differ significantly.
(d) Principal Component Analysis (PCA)
Standardize data. Compute covariance matrix. Find eigenvalues & eigenvectors.
Principal components = eigenvectors with largest eigenvalues.
Use: Reduces variables → keeps most variance.
(e) 95% Confidence Interval
Given:
n = 50, Xˉ=15\bar{X}=15Xˉ=15, s=3s=3s=3,
CI=Xˉ±zα/2sn=15±1.96×37.07CI = \bar{X} ± z_{\alpha/2}\frac{s}{\sqrt{n}} = 15 ± 1.96 \times \frac{3}{7.07}CI=Xˉ±zα/2ns=15±1.96×7.073 CI=15±0.83⇒(14.17, 15.83)CI = 15 ± 0.83 \Rightarrow (14.17,\ 15.83)CI=15±0.83⇒(14.17, 15.83)
SECTION C – Long Questions (Any 1 from each)
3(a) Monte Carlo Simulation
Generate 1000 random samples (n=30, σ=10) under H₀: μ=50.
Compute sample mean each time and check how often sample mean exceeds 95% critical limit → estimate p-value.
Reject H₀ if p < 0.05.
3(b) Markov Chains in MCMC
Markov Chain: Process where next state depends only on current state.
MCMC (e.g., Metropolis-Hastings, Gibbs sampling): draws samples from complex distributions.
Convergence: chain must reach stationary distribution for valid inference.
4(a) Monte Carlo Hypothesis Testing
Simulate sampling distribution of test statistic under H₀ to estimate p-value.
Advantages:
Works with non-normal or small samples. No strict parametric assumptions.
4(b) Jackknife Resampling
Systematically leave one observation out at a time, compute statistic θiθ_iθi, and average:
θ^jack=nθˉ−(n−1)θ−iˉ\hat{θ}_{jack} = n\bar{θ} - (n-1)\bar{θ_{-i}}θ^jack=nθˉ−(n−1)θ−iˉ
Provides estimates for bias and variance.
5(a) Permutation Test
Randomly shuffle group labels (A, B) and recompute mean difference repeatedly.
Count how often simulated |Δ| ≥ observed 2.5 → gives p-value.
6(a) 5-Fold Cross Validation
Mean Squared Errors = 12, 15, 10, 18, 14
MSEavg=12+15+10+18+145=13.8MSE_{avg} = \frac{12+15+10+18+14}{5} = 13.8MSEavg=512+15+10+18+14=13.8
6(b) History of R Language
Created by Ross Ihaka and Robert Gentleman (1993). Inspired by S language (Bell Labs).
1997: R Core Team formed. 2000: R 1.0.0 released.
Now maintained by R Foundation, used for data science, stats, and ML.
7(a) R Workspace Commands
# Save vector x <- c(2, 4, 6, 8, 10) save(x, file = "mydata.RData") # List variables in workspace ls() # Load and display values load("mydata.RData") print(x)
7(b) Vector and Matrix Creation in R
# Define vector v <- c(1, 2, 3, 4, 5) # Create 3x2 matrix from vector m <- matrix(v, nrow=3, ncol=2) # Display both print(v) print(m)