UPHESC Assistant Professor Agricultural Statistics PYQs

Concept: (Multi)collinearity = strong linear relationship among the regressors (X variables).

1. It inflates the variances of OLS coefficients (VIF = 1/(1 − R²ⱼ)) and makes signs unstable.
2. Autocorrelation/serial correlation is correlation among the error terms over time, not among regressors.

Short trick: Errors correlated ⇒ autocorrelation; X's correlated ⇒ collinearity.

Concept: A time series = trend (T) + seasonal (S) + cyclical (C) + irregular (I) components.

1. Steady long-run rise in sugar demand → trend: (a)-(3).
2. Price rise every sowing season → seasonal (within-year, recurring): (b)-(1).
3. Prosperity phase of the business cycle → cyclical: (c)-(4).
4. Strike → unpredictable, irregular: (d)-(2).

Short trick: Trend = years, Seasonal = within a year, Cycle = boom/depression, Irregular = accidents/strikes.

Concept: Heteroscedasticity = error variance changing across observations.

1. In cross-section data units differ greatly in size (small vs large farms, poor vs rich households), so the spread of errors grows with scale.
2. Time-series data more typically suffer from autocorrelation.

Short trick: Cross-section ⇒ heteroscedasticity; time series ⇒ autocorrelation.

Concept: Standard results: $R_{1.23}^2=\frac{r_{12}^2+r_{13}^2-2r_{12}{r}_{13}{r}_{23}}{1-r_{23}^2}$ and the tie-corrected Spearman formula adds $\frac{p^3-p}{12}$ to $\sum d^2$ for every group of p tied ranks.

1. In (1), $r_{12}{r}_{23}{r}_{31}$ is the same product as $r_{12}{r}_{13}{r}_{23}$ (since $r_{31}=r_{13}$), and $0\le R_{1.23}\le 1$ — correct.
2. (2) is exactly the tie-corrected Spearman coefficient — correct.

Short trick: Multiple R is never negative; ties: add (p³ − p)/12 per tie group.

Concept: Remedies for heteroscedasticity work either on the scale of the data, the standard errors, or the estimator.

1. (1) Log transformation compresses scale and often stabilises variance ✓.
2. (2) Heteroscedasticity-consistent (White's robust) standard errors ✓.
3. (3) GLS/weighted least squares re-weights observations ✓.
4. (4) Adding lagged values is a remedy for dynamic misspecification/autocorrelation, not heteroscedasticity ✗.

Short trick: Log, robust SE, GLS/WLS — the three cures.

Concept: $a=b_{yx}=r\frac{{\sigma }_y}{{\sigma }_x}$ and $c=b_{xy}=r\frac{{\sigma }_x}{{\sigma }_y}$; both lines pass through $(\overline{x},\overline{y})$.

1. $\frac{c}{a}=\frac{{\sigma }_x^2}{{\sigma }_y^2}$ — (1) true.
2. $ac=r^2$ ⇒ $r=\sqrt{ac}$ (taking the common sign of a and c) — (2) true.
3. Solve $\overline{y}=a\overline{x}+b$, $\overline{x}=c\overline{y}+d$: $\overline{x}=c(a\overline{x}+b)+d$ ⇒ $\overline{x}=\frac{cb+d}{1-ac}$, and $\overline{y}=\frac{ad+b}{1-ac}$ — (3) true.

Short trick: $b_{xy}/b_{yx}={\sigma }_x^2/{\sigma }_y^2$; $b_{xy}{b}_{yx}=r^2$; lines meet at the means.

Concept: Degeneracy = a basic variable takes the value zero in a basic feasible solution.

1. Stage 1: in the initial basic feasible solution (some bᵢ = 0).
2. Stage 2: at a subsequent iteration, when there is a tie in the minimum-ratio test for the leaving variable.
3. Degeneracy can cause cycling; it is resolved by perturbation or Bland's rule.

Short trick: Degeneracy: at the start, or during iterations — two stages.

Concept: For any real t, $E[(X-tY{)}^2]\ge 0$, i.e. $E(X^2)-2tE(XY)+t^{2}E(Y^2)\ge 0$.

1. A quadratic in t that is never negative has discriminant ≤ 0: $4[E(XY){]}^2-4E(X^2)E(Y^2)\le 0$.
2. So $[E(XY){]}^2\le E(X^2)E(Y^2)$, with equality when Y is a multiple of X (hence ≤, not strict <).
3. Applied to deviations from means it gives |r| ≤ 1.

Short trick: Cauchy–Schwarz ⇒ |r| ≤ 1.

Concept: Gradient-based optimisation uses the first derivative to locate an optimum and the second derivative to classify it.

1. (1) Direct (derivative-free) search suits non-differentiable functions ✓.
2. (2) ∇f points in the direction of steepest ascent ✓.
3. (3) The Hessian = matrix of second-order partials = Jacobian of the gradient ✓.
4. (4) Reaching an optimal point is detected by the gradient (first derivative) being zero; the second derivative only tells whether it is a maximum, minimum or saddle ✗.

Short trick: First derivative = 0 → stationary point; second derivative → what kind of point.

Concept: Standard definitions (Kanti Swarup, Taha): solution → feasible solution → basic feasible solution → optimal solution.

1. Solution: any set of values of the decision variables satisfying the constraints (2) — statement 1 true.
2. Feasible solution: a solution that also satisfies the non-negativity restrictions (3) — statement 2 true.
3. Optimal solution: a feasible solution that optimises Z.

Short trick: Constraints ⇒ solution; + non-negativity ⇒ feasible; + best Z ⇒ optimal.

Concept: Random (direct, derivative-free, black-box) search only evaluates f at trial points and keeps the better ones.

1. Because it never computes a gradient (R), it does not need f to be continuous or differentiable (A).
2. So both are true and R is the reason for A.

Short trick: No gradient needed ⇒ works on non-smooth functions.

Concept: Steepest ascent: $x_{k+1}=x_k+h^{\ast }\mathrm{\nabla }f(x_k)$, with $h^{\ast }$ chosen to maximise f along the gradient.

1. (1) The method needs ∇f, so f must be differentiable ✓.
2. (2) At any non-optimal point of the (unimodal) function the gradient is non-zero; it vanishes only at the optimum ✓.
3. (3) Each optimal line search moves uphill, so f increases from iteration to iteration ✓.

Short trick: Gradient exists, is non-zero away from the peak, and every step climbs.

Concept: If all decision variables must be integers the problem is an all (pure) IPP; if only some are, it is a mixed IPP.

1. Statement (1) describes 'some $x_j$ integer' — that is a mixed IPP, so calling it an all-IPP is false.
2. Statement (2): the Branch and Bound method was developed by A. H. Land and A. G. Doig (1960). The statement names 'A. L. Long and A. P. Doig' — read literally, the attribution is wrong and (2) is false, giving (D); read as meant for Land and Doig, (2) is true, giving (B).
3. Since the printed statement supports two different options, no single option can be marked correct.

Short trick: All integer = pure IPP; some integer = mixed IPP; B&B = Land & Doig (1960); cutting plane = Gomory (1958).

Concept: In C++ a derived class lists all its base classes after the colon, separated by commas.

1. Example: class C : public A, public B { … };
2. Dot is the member-access operator; # starts preprocessor directives; the dollar sign has no role here.

Short trick: Colon to inherit, comma to inherit from many.

Concept: A DBMS (e.g. MySQL, Oracle, MS Access) sits between users/programs and the stored data.

1. It provides CRUD operations — create, read, update, delete — plus security (protect): statement 1 true.
2. Centralised shared data with integrity constraints, concurrency control and reduced redundancy is its key advantage: statement 2 true.

Short trick: DBMS = CRUD + sharing + integrity + security.

Concept: Basic web-development toolkit: local server stack for dynamic sites, editor for pages, FTP client for upload.

1. (1) XAMPP (cross-platform/Windows) and LAMPP (Linux) bundle Apache + MySQL + PHP/Perl — a local server for dynamic pages ✓.
2. (2) WYSIWYG ('what you see is what you get') visual HTML editors help beginners build static pages ✓.
3. (3) Free FTP clients (e.g. FileZilla) upload site files to the hosting server ✓.

Short trick: Dynamic → XAMPP/LAMPP; static → WYSIWYG editor + FTP client.

Concept: Jackknife (Quenouille 1949, Tukey 1958): recompute the statistic leaving out one observation at a time; bootstrap (Efron 1979): resample n observations with replacement.

1. Jackknife estimates bias and variance and can be viewed as a linear approximation to the bootstrap — A true.
2. The bootstrap draws samples with replacement, not without replacement — R false.

Short trick: Jackknife = leave-one-out; Bootstrap = with replacement.

Concept: HTML gives structure, CSS gives style, JavaScript gives behaviour.

1. HTML is the standard markup language for web pages, supported by CSS and JS — statement 1 true.
2. A page is built from elements (tags such as <h1>, <p>, <img>) that tell the browser how to display content — statement 2 true.

Short trick: HTML = structure, CSS = look, JS = action.

Concept: A stack is a LIFO (last in, first out) structure; push inserts at the top and pop removes from the top.

1. Arrays and linked lists can be used to implement a stack, but push/pop are the stack's own operations.
2. Queues use enqueue/dequeue instead.

Short trick: Stack of plates: push on top, pop from top (LIFO).

Concept: The CPU's internal structure (Stallings): control unit, ALU, registers and the CPU interconnection.

1. Control unit — controls the operation of the CPU (and computer): (a)-(3).
2. ALU — performs the processing (arithmetic and logic) functions: (b)-(1).
3. Registers — fast storage internal to the CPU: (c)-(4).
4. CPU interconnection (internal bus) — communication among CU, ALU and registers: (d)-(2).

Short trick: CU controls, ALU computes, registers store, bus connects.