UPHESC Assistant Professor Statistics 2025 PYQs

Concept: Non-response = failure to obtain information from a selected unit, for any reason.

1. Not-at-home / non-availability, non-return of mailed questionnaires and outright refusal are all forms of non-response.
2. It is a non-sampling error; handled by call-backs, Hansen–Hurwitz sub-sampling, Politz–Simmons method.

Short trick: Any reason for 'no data from a selected unit' = non-response.

Concept: Complexity grows with the number of restrictions on randomisation / sources of variation controlled.

1. CRD: no local control – one-way ANOVA (simplest).
2. RBD: one restriction (blocks) – two-way ANOVA.
3. LSD: two restrictions (rows and columns) – three-way classification.
4. So a → b → c; BIBD (d) is more complex still, and options with d first or between a and c are wrong.

Short trick: CRD (0 blocking) → RBD (1) → LSD (2).

Concept: Almost sure ⇒ in probability ⇒ in distribution (converses fail in general).

1. The weakest mode is the one implied by all others: convergence in distribution.
2. Convergence in probability is in the middle.
3. Almost sure convergence is strongest. Order: b, a, c.

Short trick: a.s. ⇒ P ⇒ D: read it backwards for weakest → strongest.

Concept: PCA is a model-free, purely variance-based technique usable on almost any multivariate data set; factor analysis needs a latent-factor model; Mahalanobis distance is a specific distance measure for a particular purpose.

1. PCA (b): only needs a covariance/correlation matrix – most general (data reduction, visualisation, pre-processing).
2. Factor analysis (c): assumes the common-factor model $X=\mathrm{\Lambda }F+\epsilon$ – narrower than PCA.
3. Mahalanobis distance (a): a single tool for distance/classification/outlier detection – most specific.
4. Descending order: b, c, a.

Short trick: General tool → model-based tool → single-purpose measure.

Concept: A BIBD$(v,b,r,k,\lambda )$ arranges $v$ treatments in $b$ blocks of size $k$ so that every treatment occurs in $r$ blocks and every pair together in $\lambda$ blocks.

1. (A) is true: equal replication $r$ for every treatment is part of the structure.
2. (R) is true: the parameters are tied by the combinatorial relations $vr=bk$, $\lambda (v-1)=r(k-1)$, $b\ge v$.
3. These relations fix $r$ uniquely for all treatments ($r=bk/v$), which is why each treatment appears in the same number of blocks – (R) explains (A).

Short trick: BIBD identities: $vr=bk$ and $\lambda (v-1)=r(k-1)$.

Concept: Planning (factors, objective/response) comes first, then randomisation, then running the experiment.

1. Choosing factors/levels and defining the response are the two planning steps; Montgomery notes they are often done together or in reverse order.
2. Randomisation of treatments to units (IV) must precede conducting the trial (III).
3. Only option (A) has both planning steps first and IV before III: I, II, IV, III. Options with III before IV, or with planning steps at the end, are impossible.

Short trick: Plan → randomise → run.

Concept: Neyman factorisation: $T$ is sufficient if $L(\theta )=g(T,\theta )h(x)$.

1. $L=\prod _{k=1}^{3}k\theta e^{-k\theta x_k}=6{\theta }^3{e}^{-\theta (x_1+2x_2+3x_3)}$.
2. $\theta$ enters only through $x_1+2x_2+3x_3$.
3. So $T=X_1+2X_2+3X_3$ is sufficient (the rates are $\theta ,2\theta ,3\theta$, so weights are 1, 2, 3).

Short trick: Weight each $X_k$ by its own rate multiplier k.

Concept: ${\sigma }^2=\frac{\sum x^2}{n}-{\overline{x}}^2$.

1. The values are symmetric about 0, so $\overline{x}=0$.
2. $\sum x^2=2(1+2+\cdots +7)=2\times 28=56$, $n=14$.
3. ${\sigma }^2=56/14=4$, $\sigma =2$.

Short trick: Symmetric ± pairs: mean 0, S.D. = $\sqrt{\text{mean of squares}}$.

Concept: ${\beta }_2={\mu }_4/{\mu }_2^2$.

1. $X-\mu =\pm \sigma$, each with probability ½.
2. ${\mu }_2={\sigma }^2$, ${\mu }_4={\sigma }^4$.
3. ${\beta }_2={\sigma }^4/{\sigma }^4=1$ – the minimum possible value of ${\beta }_2$ (two-point symmetric distribution).

Short trick: ${\beta }_2\ge 1$ always; equality only for a symmetric two-point law.

Concept: Logistic model: $\frac{dN}{dt}=rN(1-\frac{N}{K})$.

1. As $N\to K$, the factor $1-\frac{N}{K}\to 0$, so $\frac{dN}{dt}\to 0$.
2. Growth is fastest at $N=K/2$ and levels off at the carrying capacity.
3. It becomes negative only if $N>K$.

Short trick: S-curve flattens at K: growth rate → 0.

Concept: Bartlett: ${\chi }_0^2=\frac{q}{c}$ with natural logs; with common logs ${\chi }_0^2=2.3026\frac{q}{c}$.

1. $q=(N-a){\log }_{10}{S}_p^2-\sum (n_i-1){\log }_{10}{S}_i^2$.
2. $c=1+\frac{1}{3(a-1)}[\sum \frac{1}{n_i-1}-\frac{1}{N-a}]$.
3. $2.3026=\ln 10$ converts ${\log }_{10}$ to $\ln$; ${\chi }_0^2\sim {\chi }_{a-1}^2$ under $H_0$.

Short trick: 2.3026 = ln 10 – the usual base-change constant.

Concept: LTPD (Lot Tolerance Percent Defective) $p_t$ is the worst lot quality the consumer is willing to tolerate.

1. AQL = quality level that is acceptable (satisfactory) – better than LTPD.
2. The probability of accepting a lot of quality $p_t$ is the consumer's risk $\beta$.
3. AOQL = maximum average outgoing quality after rectification; producer's risk = rejecting a good (AQL) lot.

Short trick: AQL ↔ producer's risk α; LTPD ↔ consumer's risk β.

Concept: $\text{Var}({\hat{\beta }}_j)=\frac{{\sigma }^2}{S_{jj}}\cdot \frac{1}{1-R_j^2}$; the factor $\frac{1}{1-R_j^2}$ is the variance inflation factor.

1. If $X_j$ is unrelated to the other regressors, $R_j^2=0$ and VIF = 1.
2. As $R_j^2\to 1$ (strong collinearity), VIF $\to \mathrm{\infty }$; VIF > 10 is the usual warning sign.
3. The square root (option C) inflates the standard error, not the variance.

Short trick: VIF = 1/tolerance, tolerance = $1-R_j^2$.

Concept: In partial confounding of a $2^3$ design with 4 replications, each interaction (AB, AC, BC, ABC) is confounded in one replicate only.

1. Each partially confounded effect is estimated from the 3 replicates in which it is not confounded.
2. Relative information = (replicates unconfounded)/(total replicates) = $\frac{3}{4}$.
3. Yates (1937) called this ratio the relative information of the confounded effects.

Short trick: Confounded in 1 of 4 replicates → information 3/4.

Concept: Regression coefficients are independent of change of origin but NOT of scale.

1. If $u=\frac{x-a}{h}$, $v=\frac{y-b}{k}$ then $b_{yx}=\frac{k}{h}{b}_{vu}$ – depends on the scales $h,k$. So (A) is false.
2. $r=\pm \sqrt{b_{yx}{b}_{xy}}$ – $r$ is the geometric mean of the two regression coefficients. So (R) is true.
3. (It is $r$, not $b$, that is independent of both origin and scale.)

Short trick: r: origin ✗ scale ✗ (independent of both); b: origin ✗, scale ✓ (depends).

Concept: $D^2=(x-\mu {)}^{\prime }{\mathrm{\Sigma }}^{-1}(x-\mu )$ measures how far a point lies from the centre of a multivariate distribution, allowing for variances and correlations.

1. It generalises the standardised distance $\frac{(x-\mu {)}^2}{{\sigma }^2}$ to many variables.
2. Uses: classification/discriminant analysis, outlier detection, distance between populations (P.C. Mahalanobis, 1936).
3. Testing variances (Bartlett/Box M) and means (Hotelling $T^2$) are other procedures; $T^2$ merely uses a $D^2$-type form.

Short trick: D² = scale-free, correlation-adjusted distance from a distribution.

Concept: Median $M$ satisfies $F(M)=\frac{1}{2}$; for $f(x)=\lambda e^{-\lambda x}$, $F(x)=1-e^{-\lambda x}$.

1. $1-e^{-2M}=\frac{1}{2}$ ⇒ $e^{-2M}=0.5$.
2. $-2M=\ln 0.5$ ⇒ $M=-\frac{1}{2}\ln 0.5=\frac{\ln 2}{2}\approx 0.347$.
3. ½ is the mean, not the median.

Short trick: Exponential median = $\frac{\ln 2}{\lambda }$ < mean $\frac{1}{\lambda }$.

Concept: Each graph has a characteristic use.

1. Box plot – displays the five-number summary and flags outliers (a–IV).
2. Ogive – cumulative frequency curve used to locate the median $Q_2$ graphically (b–I).
3. Histogram – for a continuous quantitative variable (c–II).
4. Bar chart – for a categorical variable (d–III).

Short trick: Box→outliers, Ogive→median, Histogram→numbers, Bar→categories.

Concept: Standard procedure of a significance test.

1. Set up $H_0$ and $H_1$ (I).
2. Fix the level of significance $\alpha$ before seeing the data (IV).
3. Compute the test statistic from the sample (III).
4. Compare with the critical value / p-value and decide (II).

Short trick: Hypothesis → α → statistic → decision.

Concept: Probability that a person aged 20 survives to age 25: ${}_5{p}_{20}=\frac{l_{25}}{l_{20}}$.

1. $\frac{l_{25}}{l_{20}}=\frac{93393}{95510}=0.9778\approx 0.98$.
2. 0.02 is the complementary probability of dying ${}_5{q}_{20}$.
3. Other ratios (e.g. $l_{30}/l_{20}=0.944$) do not correspond to the 20 → 25 interval.

Short trick: p = l(later)/l(earlier); q = 1 − p.