UGC NET Environmental Science Unit-IX: Statistical Approaches and Modelling in Environmental Sciences Syllabus

Differentiate between mutually exclusive and independent events.

Mutually exclusive events cannot occur together (P(A∩B)=0). Independent events are those whose occurrence does not affect each other (P(A∩B)=P(A)P(B)). Mutually exclusive events with non-zero probabilities are necessarily dependent.

What is the range of probability values, and what do 0 and 1 mean?

Probability ranges from 0 to 1. P=0 means an impossible event; P=1 means a certain event.

Differentiate between a population (universe) and a sample.

A population is the entire group/totality of items under study; a sample is a representative subset of the population selected for study to draw inferences about the population.

Distinguish between a parameter and a statistic.

A parameter is a numerical value describing a characteristic of a population (e.g., μ, σ). A statistic is a value computed from a sample (e.g., x̄, s) used to estimate the parameter.

What is the main difference between probability (random) sampling and non-probability sampling?

In probability sampling every unit has a known, non-zero chance of selection (allows estimation of sampling error); in non-probability sampling selection is based on judgement/convenience and selection probabilities are unknown.

Name four common probability sampling methods.

Simple random sampling, Stratified random sampling, Systematic sampling, and Cluster (and multistage) sampling.

What is stratified random sampling and when is it preferred?

The population is divided into homogeneous subgroups (strata) and random samples are drawn from each stratum. It is preferred when the population is heterogeneous, improving representativeness and precision.

What is sampling error?

Sampling error is the difference between a sample statistic and the true population parameter, arising because only a part of the population is studied. It decreases as sample size increases.

State the Central Limit Theorem (CLT).

The sampling distribution of the sample mean approaches a normal distribution as sample size (n) increases, regardless of the population's distribution, with mean μ and standard error σ/√n (generally n ≥ 30).

What is the standard error of the mean and its formula?

The standard error (SE) is the standard deviation of the sampling distribution of the mean: SE = σ/√n. It decreases as sample size increases.

Differentiate between Type I and Type II errors in hypothesis testing.

Type I error (α): rejecting a true null hypothesis (false positive). Type II error (β): accepting a false null hypothesis (false negative).

What is the null hypothesis (H₀)?

The null hypothesis is a statement of no difference or no effect/relationship, assumed true until evidence rejects it; it is the hypothesis actually tested.

List the key properties of the normal distribution.

Bell-shaped, symmetric about the mean; mean = median = mode; total area = 1; asymptotic to the x-axis; defined by mean (μ) and SD (σ); follows the empirical 68-95-99.7% rule.

State the empirical (68-95-99.7) rule for a normal distribution.

About 68% of data lie within μ ± 1σ, ~95% within μ ± 2σ, and ~99.7% within μ ± 3σ.

What is the standard normal distribution and the Z-score formula?

A normal distribution with mean 0 and SD 1. The Z-score standardizes a value: Z = (x − μ)/σ, giving deviation from the mean in SD units.

Describe the binomial distribution and its mean and variance.

A discrete distribution for the number of successes in n independent trials each with success probability p. Mean = np; Variance = npq (where q = 1−p).

Describe the Poisson distribution and state its key property.

A discrete distribution for the number of rare events in a fixed interval of time/space, with parameter λ. Its key property is that mean = variance = λ.

Under what condition does the binomial distribution approximate the Poisson distribution?

When n is large and p is very small (rare events), with np = λ remaining moderate/finite.

Define correlation and the range of the correlation coefficient.

Correlation measures the degree and direction of the linear relationship between two variables. The coefficient (r) ranges from −1 (perfect negative) through 0 (no linear correlation) to +1 (perfect positive).

What is Karl Pearson's coefficient of correlation formula?

r = Σ[(x−x̄)(y−ȳ)] / [√Σ(x−x̄)² · √Σ(y−ȳ)²], i.e., covariance(x,y)/(σx·σy).

What is Spearman's rank correlation coefficient and when is it used?

ρ = 1 − [6Σd² / n(n²−1)], where d is the difference between paired ranks. Used for ranked/ordinal data or non-linear monotonic relationships.

What is the coefficient of determination (r²) and what does it indicate?

r² is the square of the correlation coefficient; it gives the proportion of variation in the dependent variable explained by the independent variable (ranges 0 to 1).

Differentiate between correlation and regression.

Correlation measures the strength/direction of association between two variables (symmetric, no cause-effect). Regression estimates/predicts the value of a dependent variable from an independent variable using a mathematical equation (asymmetric).

What is the simple linear regression equation of Y on X?

Y = a + bX, where 'a' is the intercept and 'b' is the regression coefficient (slope) = r·(σy/σx), estimated by least squares to minimize the sum of squared residuals.

What is the relationship between the two regression coefficients and the correlation coefficient?

r = ±√(byx · bxy); the correlation coefficient is the geometric mean of the two regression coefficients, and r takes the common sign of both.

What are the main approaches to the development of environmental models?

Empirical (statistical/black-box, data-driven) models and Mechanistic (deterministic/process-based) models; models may also be classified as deterministic vs stochastic, static vs dynamic, and steady-state vs dynamic.

Differentiate between deterministic and stochastic environmental models.

A deterministic model gives a fixed output for given inputs with no randomness (same input → same output). A stochastic model incorporates randomness/probability, so outputs are expressed as probability distributions.

List the main steps in developing an environmental model.

Problem definition/conceptualization, formulation (mathematical equations), parameter estimation, calibration, verification, validation, sensitivity analysis, and application/prediction.

What is the difference between calibration and validation of a model?

Calibration adjusts model parameters so outputs fit a known/observed dataset. Validation tests the calibrated model against an independent dataset to confirm its predictive reliability.

Describe the exponential (Malthusian) model of population growth and its equation.

Unlimited growth at a constant per-capita rate giving a J-shaped curve: dN/dt = rN, with solution Nₜ = N₀·e^(rt), where r is the intrinsic rate of natural increase.

Describe the logistic model of population growth, its equation, and curve shape.

Density-dependent growth limited by carrying capacity (K), giving an S-shaped (sigmoid) curve: dN/dt = rN[(K−N)/K]. Growth is fastest at N = K/2 and stops at N = K (Verhulst-Pearl equation).

What is carrying capacity (K) in population ecology?

Carrying capacity is the maximum population size of a species that an environment can sustain indefinitely given available resources; population growth rate becomes zero when N = K.

State the Lotka-Volterra predator-prey model equations.

Prey: dN/dt = aN − bNP; Predator: dP/dt = cbNP − dP, where N=prey, P=predator. It produces coupled, out-of-phase oscillations in predator and prey populations.