Mathematical and Statistical Foundations

Introduction

Data analytics rests on mathematical and statistical reasoning. Software can calculate a mean, fit a model, or generate a probability, but it cannot determine whether the result is appropriate, whether assumptions are credible, or whether an observed pattern is practically meaningful. Postgraduate analytics therefore requires both computational competence and judgement about uncertainty, evidence, measurement, and generalisation.

This chapter develops the foundations needed to analyse data responsibly. It covers probability, random variables and distributions, sampling, estimation, hypothesis testing, correlation, regression, uncertainty measurement, Bayesian reasoning, and matrix concepts. The emphasis is on interpretation and application rather than mechanical calculation. These ideas support later work in machine learning, experiment design, forecasting, optimisation, and decision analytics.

Learning Objectives

By the end of this chapter, students should be able to:

1. Explain probability as a model of uncertainty and distinguish conditional, joint, and marginal probability.

2. Use common discrete and continuous probability distributions appropriately.

3. Explain sampling variation, bias, the central limit theorem, and the law of large numbers.

4. Construct and interpret point estimates and confidence intervals.

5. Formulate and evaluate null and alternative hypotheses while distinguishing statistical from practical significance.

6. Interpret covariance, correlation, and the foundations of linear regression.

7. Quantify uncertainty and explain the difference between frequentist and Bayesian reasoning.

8. Interpret vectors, matrices, matrix multiplication, inverses, rank, and positive definiteness in analytics.

9. Identify assumptions, limitations, ethical risks, and governance requirements in quantitative analysis.

1. Probability as a Language of Uncertainty

A probability model describes uncertain outcomes. The sample space, denoted by Ω, is the set of possible outcomes. An event A is a subset of Ω. A probability measure P assigns a number between 0 and 1 to events and satisfies three axioms:

• P(A) ≥ 0;
• P(Ω) = 1;
• for mutually exclusive events A₁, A₂, …, P(∪Aᵢ) = ΣP(Aᵢ).

The complement rule is P(Aᶜ) = 1 − P(A). If A and B are events, their union means at least one occurs, and their intersection means both occur. The addition rule is

P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

The subtraction term prevents double-counting outcomes in both events.

Conditional and joint probability

Conditional probability measures the chance of A given that B has occurred:

P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0.

Rearranging gives the multiplication rule:

P(A ∩ B) = P(A|B)P(B) = P(B|A)P(A).

Events are independent when P(A|B) = P(A), equivalently P(A ∩ B) = P(A)P(B). Independence is a strong assumption and should not be inferred merely because two variables appear unrelated in a small dataset.

Bayes’ theorem follows directly from the multiplication rule:

P(A|B) = [P(B|A)P(A)]/P(B).

For mutually exclusive and exhaustive hypotheses A₁, …, Aₖ,

P(Aᵢ|B) = [P(B|Aᵢ)P(Aᵢ)] / Σⱼ P(B|Aⱼ)P(Aⱼ).

The prior probability P(Aᵢ) is updated by evidence B through the likelihood P(B|Aᵢ), producing the posterior probability P(Aᵢ|B). A frequent error is to confuse P(A|B) with P(B|A); these quantities are generally different.

Worked example: diagnostic evidence

Suppose a condition affects 1% of a population. A test has sensitivity 90%, so P(positive|condition) = 0.90, and specificity 95%, so P(positive|no condition) = 0.05. Then

P(condition|positive) = (0.90 × 0.01)/[(0.90 × 0.01) + (0.05 × 0.99)]

= 0.009/(0.0585) ≈ 0.1538.

Thus, despite a positive test, the probability of the condition is approximately 15.4%. The result reflects the low base rate and illustrates why predictive value cannot be inferred from sensitivity alone.

2. Random Variables and Probability Distributions

A random variable assigns a numerical value to each outcome. A discrete random variable takes countable values; a continuous random variable can take values in intervals. Its probability distribution describes how probability is allocated.

For a discrete variable X, the probability mass function (PMF) is p(x) = P(X = x), with Σₓp(x) = 1. For a continuous variable, the probability density function (PDF) f(x) satisfies f(x) ≥ 0 and ∫f(x)dx = 1. A density is not itself a probability; probabilities are areas under the density. The cumulative distribution function (CDF) is F(x) = P(X ≤ x), applicable to both discrete and continuous variables.

The expected value is the probability-weighted long-run average:

E[X] = Σₓxp(x) for discrete X, and E[X] = ∫xf(x)dx for continuous X.

Variance measures dispersion around the mean μ = E[X]:

Var(X) = E[(X − μ)²] = E[X²] − μ².

The standard deviation is σ = √Var(X), expressed in the same units as X. For constants a and b, E[aX + b] = aE[X] + b and Var(aX + b) = a²Var(X). If X and Y are independent, Var(X + Y) = Var(X) + Var(Y); without independence, covariance must also be included.

Important distributions

Bernoulli distribution. A Bernoulli variable records one trial with success probability p: X = 1 with probability p and X = 0 otherwise. Its mean is p and variance is p(1 − p). It is suitable for binary outcomes under a defined trial structure.

Binomial distribution. If X counts successes in n independent Bernoulli trials with constant success probability p, then X ~ Binomial(n,p), with

P(X = k) = C(n,k)pᵏ(1 − p)ⁿ⁻ᵏ,

where k = 0, …, n. Its mean is np and variance is np(1 − p). Independence and constant p are essential assumptions.

Poisson distribution. A Poisson variable counts events in a fixed interval when events occur independently at a constant average rate λ. Its probability is

P(X = k) = e⁻λ λᵏ/k!,

and both mean and variance equal λ. In practice, overdispersion, changing rates, or dependence may invalidate the model.

Normal distribution. A normal variable has density

f(x) = [1/(σ√(2π))] exp[−(x − μ)²/(2σ²)],

where μ is the mean and σ > 0 is the standard deviation. The standardised value is z = (x − μ)/σ. The normal model is symmetric and continuous. It is often useful for measurement errors and sample means, but normality should not be assumed for heavily skewed, bounded, or count data.

Exponential distribution. For waiting times with a constant event rate λ > 0, f(x) = λe⁻λˣ for x ≥ 0. Its mean and standard deviation are both 1/λ. The memoryless property is specific to the exponential distribution and should not be applied to arbitrary waiting times.

Law of large numbers and central limit theorem

The law of large numbers states, under suitable regularity conditions such as independent identically distributed observations with finite expectation, that the sample mean converges to the population mean as sample size increases. It does not guarantee that a small sample is representative.

The central limit theorem (CLT) states, broadly, that the standardised sample mean approaches a normal distribution as sample size grows when observations are independent and identically distributed with finite variance. If X₁, …, Xₙ have mean μ and variance σ², then the sample mean X̄ has mean μ and standard error σ/√n, and its approximate distribution is normal for sufficiently large n. The required sample size depends on skewness, tail behaviour, dependence, and the intended accuracy.

3. Sampling and Data Quality

A population is the complete group of interest; a sample is the observed subset. A parameter describes a population, while a statistic describes a sample. Sampling error is random variation caused by observing a sample rather than the entire population. Sampling bias is systematic distortion introduced by the selection process or nonresponse.

Probability sampling methods include simple random sampling, systematic sampling, stratified sampling, and cluster sampling. Stratification can improve precision when strata are internally similar and differ from one another. Cluster sampling may reduce cost but often increases variance because observations within clusters resemble one another.

Non-probability methods, such as convenience, quota, and voluntary-response sampling, can be useful operationally but do not generally support design-based population inference without additional assumptions. Large sample size cannot repair severe selection bias. A million observations from a systematically unrepresentative source may be less informative than a carefully designed smaller sample.

Data quality also includes missingness, measurement error, duplicate records, inconsistent definitions, and changes in data-generating processes. Missing data may be missing completely at random, missing at random conditional on observed information, or not missing at random. The mechanism affects whether complete-case analysis, imputation, weighting, or sensitivity analysis is defensible.

4. Estimation and Uncertainty

An estimator is a rule for using sample data to estimate a parameter. An estimate is the numerical result. Desirable properties include:

• Unbiasedness: E[θ̂] = θ, where θ is the target parameter.
• Consistency: θ̂ approaches θ as sample size increases.
• Efficiency: among estimators with relevant properties, an estimator has relatively small variance.
• Robustness: performance is not excessively damaged by realistic departures from assumptions.

For observations X₁, …, Xₙ, the sample mean is X̄ = (1/n)ΣXᵢ. The sample variance is

s² = [1/(n − 1)]Σ(Xᵢ − X̄)².

The n − 1 denominator provides an unbiased estimator of the population variance under independent sampling with finite variance. The standard error of X̄ is s/√n when s estimates σ.

A confidence interval is an interval-generating procedure with a coverage property. A 95% confidence interval means that, over repeated samples and under the procedure’s assumptions, 95% of similarly constructed intervals would contain the fixed parameter. It does not mean there is a 95% probability that the fixed parameter lies in this particular interval under the classical interpretation.

For a large sample mean, an approximate 95% interval is

X̄ ± 1.96(s/√n),

when a normal approximation is credible. For a small sample from a normal population with unknown σ, the t distribution is used:

X̄ ± t₀.₉₇₅, n−1(s/√n).

The t critical value depends on degrees of freedom and is larger than 1.96 for finite samples. Confidence intervals for proportions, rates, and regression coefficients require methods appropriate to their sampling distributions; blindly applying a normal formula can yield poor coverage.

Worked example: mean estimation

A sample of 100 service times has mean 42 minutes and standard deviation 10 minutes. The standard error is 10/√100 = 1 minute. Using a large-sample 95% approximation, the interval is 42 ± 1.96(1), or (40.04, 43.96) minutes. This describes uncertainty in the population mean under the sampling and approximation assumptions. It does not state that 95% of individual service times fall in this interval; individual variation is much larger.

5. Hypothesis Testing and Decision Errors

A hypothesis test compares a null hypothesis H₀ with an alternative H₁. The null usually represents a benchmark such as no difference or no association. A test statistic measures how discrepant the observed data are from H₀. The p-value is the probability, assuming H₀ and the test assumptions are true, of obtaining a result at least as extreme as the observed result in the specified direction.

A p-value is not the probability that H₀ is true, nor is it the probability that the result occurred by chance. Statistical significance depends on sample size, variability, effect size, and the chosen significance level α. A small p-value may accompany a trivial effect in a large dataset, while a meaningful effect may be statistically inconclusive in a small study.

A Type I error rejects a true H₀; its long-run rate is controlled by α under the test procedure. A Type II error fails to reject a false H₀; its probability is β. Statistical power is 1 − β and depends on effect size, sample size, variability, α, and the alternative situation. “Fail to reject” does not prove equality.

Multiple testing increases the chance of false discoveries. If many hypotheses are tested, analysts may control the family-wise error rate, for example through conservative procedures, or control the false discovery rate, such as with the Benjamini–Hochberg approach. Pre-specification, transparent reporting, replication, and correction for multiplicity reduce selective inference.

Hypothesis testing should be connected to a decision context. Confidence intervals, effect sizes, domain thresholds, and cost-sensitive consequences generally provide more useful evidence than a binary significant/not-significant label.

6. Covariance, Correlation, and Regression Foundations

For variables X and Y, covariance is

Cov(X,Y) = E[(X − μX)(Y − μY)].

Positive covariance indicates that values above the mean of X tend to accompany values above the mean of Y. Covariance depends on measurement units. Pearson correlation standardises covariance:

ρXY = Cov(X,Y)/(σXσY),

with sample analogue r. For nonconstant variables, −1 ≤ r ≤ 1. Correlation measures linear association, not causation. It can be distorted by outliers, restriction of range, nonlinear relationships, aggregation, and confounding.

The simple linear regression model is

Yᵢ = β₀ + β₁Xᵢ + εᵢ,

where β₀ is the intercept, β₁ is the expected change in Y for a one-unit increase in X, and εᵢ is an error term. Ordinary least squares estimates coefficients by minimising Σ(Yᵢ − b₀ − b₁Xᵢ)². With an intercept and nonzero variation in X,

b₁ = Σ(Xᵢ − X̄)(Yᵢ − Ȳ) / Σ(Xᵢ − X̄)²,

and b₀ = Ȳ − b₁X̄.

The coefficient of determination R² is the proportion of observed variation in Y explained by the fitted linear model relative to an intercept-only model:

R² = 1 − SSE/SST,

where SSE = Σ(Yᵢ − Ŷᵢ)² and SST = Σ(Yᵢ − Ȳ)². In simple regression with an intercept, R² = r², but a high R² does not establish causation or guarantee useful predictions outside the observed range.

Classical inference for regression commonly assumes independent errors, correct functional form, constant error variance, and approximately normal errors for small-sample coefficient tests. Diagnostics should examine residual patterns, leverage, influential observations, heteroscedasticity, and dependence. Transformations, weighted least squares, robust standard errors, nonlinear models, or resampling may be more suitable when assumptions fail.

Multiple regression extends the model to Y = β₀ + β₁X₁ + … + βₚXₚ + ε. A coefficient is interpreted conditionally: βⱼ describes the expected change associated with Xⱼ holding the included predictors fixed, subject to model validity. High correlation among predictors produces multicollinearity, which can make individual coefficients unstable even when predictions remain adequate. Regularisation and careful feature design can help, but they do not automatically solve confounding or measurement bias.

7. Bayesian Reasoning and Uncertainty Measurement

Frequentist inference treats parameters as fixed but unknown and evaluates the long-run behaviour of procedures. Bayesian inference represents uncertainty about parameters with a prior distribution and updates it using data:

p(θ|D) = p(D|θ)p(θ)/p(D),

where θ is a parameter, D is data, p(θ) is the prior, p(D|θ) is the likelihood, p(θ|D) is the posterior, and p(D) is the evidence or normalising constant.

A posterior distribution supports credible intervals, posterior probabilities, and predictive distributions. A 95% credible interval can be interpreted, within the model, as containing 95% posterior probability. This differs from a confidence interval. Bayesian conclusions depend on the prior and likelihood; prior sensitivity should be investigated, especially when data are limited or the prior is influential.

Uncertainty has several sources: random sampling variation, measurement error, model uncertainty, parameter uncertainty, data shift, and unobserved confounding. A single standard error does not capture all of them. Sensitivity analysis, bootstrap resampling, simulation, scenario analysis, prediction intervals, and Bayesian posterior predictive checks are useful tools.

A prediction interval is wider than a confidence interval for a mean because it includes both uncertainty in the estimated mean and individual-level variability. Communicating uncertainty should identify the quantity being estimated, the population and time period, assumptions, interval type, and important exclusions.

8. Matrix Concepts for Analytics

A vector is an ordered collection of numbers; a matrix is a rectangular array. In data analytics, a dataset with n observations and p features is commonly represented by an n × p design matrix X. A coefficient vector β has p components, and predictions in a linear model can be written as ŷ = Xβ, where matrix multiplication is defined because the inner dimensions agree.

For matrices A and B, (AB)ᵢⱼ = ΣₖAᵢₖBₖⱼ. Matrix multiplication is generally not commutative: AB need not equal BA. The transpose Xᵀ exchanges rows and columns. A square matrix A has an inverse A⁻¹ when it is nonsingular, satisfying A⁻¹A = I. Inversion is not defined for every matrix and is numerically less stable than solving a linear system directly.

The ordinary least-squares objective is ||y − Xβ||². If X has full column rank, differentiating the objective yields the normal equations XᵀXβ = Xᵀy and the formal solution

β̂ = (XᵀX)⁻¹Xᵀy.

The full-rank condition is important. In practice, QR or singular value decomposition is preferred to explicitly calculating (XᵀX)⁻¹ because it is more numerically stable, especially under multicollinearity.

A symmetric matrix satisfies A = Aᵀ. A covariance matrix is symmetric and, theoretically, positive semidefinite: for every vector z, zᵀΣz ≥ 0. Positive definiteness, zᵀAz > 0 for every nonzero z, supports unique quadratic minima and reliable optimisation. Eigenvalues and eigenvectors describe directions of transformation; principal component analysis uses eigenvectors of a covariance or correlation matrix to identify orthogonal directions of high variance.

Scaling matters in matrix-based analytics. Features measured in large units can dominate distance calculations or regularisation unless standardisation is appropriate. Singular matrices may result from redundant features, exact linear dependence, or inadequate data variation. These are not merely computational nuisances; they often signal an identification or design problem.

9. Illustrative Applied Case: Predicting Customer Support Resolution Time

Background. An online service provider wants to understand why support tickets take different amounts of time to resolve and improve staffing decisions. The analytics team receives 8,000 historical tickets containing category, priority, customer region, number of previous contacts, and resolution time.

Problem. Management proposes using a linear regression model and asks whether high-priority tickets are resolved faster. The data contain missing region values, repeated tickets from the same customers, and a policy change midway through the observation period.

Analysis. The team first defines the target precisely: elapsed hours from ticket creation to final resolution, excluding automatically closed tickets. It examines the distribution and finds strong right skew, so the median and upper percentiles are reported in addition to the mean. Repeated customers mean observations are not fully independent. The policy change creates possible time-period confounding, and priority may be assigned in response to the perceived difficulty of a ticket.

The team stratifies descriptive summaries by period and ticket category, examines missingness, and creates a log-transformed outcome only after documenting the interpretation. A regression model includes priority, category, period, and prior-contact count, with robust standard errors and customer-aware validation. Residual plots reveal changing variance, and a few high-leverage enterprise tickets are investigated rather than silently deleted. A sensitivity analysis compares complete-case results with a missing-region indicator and multiple-imputation approach.

Approach and outcome. The analysis finds that high-priority tickets have shorter median resolution times after adjustment, but the association is not interpreted causally because priority is assigned non-randomly. The strongest operational predictor is prior-contact count, while the policy change materially affects all resolution times. Management uses prediction intervals for staffing scenarios rather than treating point predictions as guarantees. A dashboard displays sample size, period, uncertainty intervals, data freshness, and known exclusions.

Learning points. The case demonstrates that statistical foundations govern the entire analytical lifecycle: defining variables, recognising dependence, checking distributions, modelling uncertainty, validating predictions, and separating association from causation. It also shows why governance requires documentation of transformations, missing-data decisions, access controls, and limitations.

10. Professional Practice, Ethics, and Contemporary Developments

Analytical results can affect credit, employment, healthcare, education, and public services. Responsible practice requires lawful data use, purpose limitation, data minimisation, security, documentation, and proportionate retention. Analysts should evaluate whether variables act as proxies for protected characteristics, whether sampling excludes vulnerable groups, and whether error rates differ across populations.

Reproducibility is strengthened by version-controlled code, frozen data extracts or documented data lineage, preregistered analysis plans where feasible, clear variable definitions, automated tests, and independent review. Privacy-preserving techniques such as aggregation, pseudonymisation, differential privacy, and secure computation can reduce disclosure risk, but none is a universal guarantee. Re-identification risk may remain when multiple datasets are linked.

Contemporary analytics increasingly uses probabilistic programming, causal inference, bootstrap and permutation methods, high-dimensional regularisation, and simulation-based calibration. AI systems may automate estimation or prediction, but their outputs still depend on data quality, sampling, objective definition, and assumptions. Calibration, uncertainty quantification, drift monitoring, and human oversight are especially important when models are deployed in changing environments.

Best practice is to begin with the decision and estimand, not the available algorithm. State the population, outcome, time horizon, comparison, assumptions, and acceptable error. Use exploratory analysis to discover structure, confirmatory analysis to test pre-specified claims, and sensitivity analysis to assess fragility. Report effect sizes and uncertainty, not only p-values or rankings.

Chapter Summary

Probability provides the formal language for uncertain events, while conditional probability and Bayes’ theorem explain how evidence updates beliefs. Random variables and distributions model outcomes; expectation, variance, and standard deviation quantify location and dispersion. The law of large numbers and central limit theorem explain why sampling and averages support inference, but only under appropriate conditions.

Sampling design determines whether conclusions can generalise. Estimation uses sample information to learn about population parameters, and confidence intervals describe long-run coverage. Hypothesis testing must be interpreted through effect size, power, multiple testing, and practical consequences. Covariance and correlation describe association, while regression models conditional relationships under explicit assumptions; neither automatically establishes causation.

Bayesian analysis represents uncertainty through prior and posterior distributions, whereas frequentist procedures emphasise long-run operating characteristics. Matrix algebra provides the language for regression, covariance, optimisation, and dimensionality reduction. Across all methods, uncertainty, data quality, dependence, model assumptions, fairness, privacy, and reproducibility are central professional responsibilities.

Key Terms

Probability: A numerical measure of uncertainty satisfying the probability axioms.

Conditional probability: The probability of an event given that another event has occurred.

Bayes’ theorem: A rule for updating a probability using prior information and evidence.

Random variable: A function assigning numerical values to outcomes.

Probability mass function: Probabilities assigned to values of a discrete random variable.

Probability density function: A function whose area over an interval gives the probability for a continuous variable.

Expected value: The probability-weighted average or long-run mean of a random variable.

Variance: The expected squared deviation from a variable’s mean.

Sampling error: Random variation resulting from observing a sample rather than a population.

Sampling bias: Systematic distortion caused by the sampling or measurement process.

Estimator: A rule that uses sample data to estimate a population parameter.

Confidence interval: An interval produced by a procedure with a specified long-run coverage property.

Null hypothesis: A benchmark claim evaluated by a statistical test.

p-value: The probability, under the null and assumptions, of data at least as extreme as observed.

Statistical power: The probability of rejecting a false null hypothesis under specified conditions.

Covariance: A measure of joint variation between two variables.

Correlation: Standardised covariance measuring linear association.

Regression: A model for relating an outcome to one or more predictors.

Posterior distribution: Updated uncertainty about a parameter after observing data.

Design matrix: A matrix containing observations of predictors used in a model.

Review Questions

Short Answer Questions

1. Distinguish between conditional probability and joint probability.

2. What assumptions are required for a binomial distribution?

3. Explain why a confidence interval is not a probability statement about a fixed parameter in the classical framework.

4. Differentiate a Type I error from a Type II error.

5. Why does correlation not establish causation?

Long Answer Questions

1. Explain the central limit theorem and discuss when a normal approximation to a sample mean may be unreliable.

2. Compare frequentist confidence intervals with Bayesian credible intervals, including their interpretations and assumptions.

3. Describe the major sources of sampling bias and propose methods for reducing them in an organisational survey.

4. Explain ordinary least squares regression, including coefficient interpretation, R², residual diagnostics, and multicollinearity.

5. Discuss how matrix algebra supports regression and principal component analysis.

Analytical / Application Questions

1. A fraud alert system flags 2% of transactions. It detects 95% of fraudulent transactions and falsely flags 3% of legitimate transactions. Calculate the probability that a flagged transaction is actually fraudulent and explain the base-rate implication.

2. A sample of 64 observations has mean 18 and standard deviation 8. Construct an approximate 95% confidence interval for the population mean and state the assumptions required.

3. A model shows a statistically significant association between customer income and loan approval. Design an analysis to assess whether the association is causal, identify potential confounders, and address fairness and governance risks.

Multiple Choice Questions with Answers

Question 1

A disease has prevalence 1%, sensitivity 90%, and false-positive rate 5%. What is approximately P(disease | positive test)?

A. 1.0%

B. 5.0%

C. 15.4%

D. 90.0%

Correct Answer: C. 15.4%

Question 2

If X ~ Binomial(20, 0.30), what is E[X]?

A. 0.30

B. 6

C. 14

D. 20.30

Correct Answer: B. 6

Question 3

What does a 95% classical confidence interval mean?

A. There is a 95% probability that the fixed parameter is in this interval

B. 95% of individual observations lie in this interval

C. The procedure captures the parameter in 95% of repeated samples, under its assumptions

D. The null hypothesis has a 95% probability of being true

Correct Answer: C. The procedure captures the parameter in 95% of repeated samples, under its assumptions

Question 4

If two events A and B are independent, which relationship holds?

A. P(A ∩ B) = P(A) + P(B)

B. P(A | B) = P(B | A) always

C. P(A ∩ B) = P(A)P(B)

D. P(A ∪ B) = 0

Correct Answer: C. P(A ∩ B) = P(A)P(B)

Question 5

A p-value is best described as:

A. The probability that the null hypothesis is true

B. The probability of data at least as extreme as observed, assuming the null and model assumptions

C. The probability that the alternative hypothesis is false

D. The practical size of an effect

Correct Answer: B. The probability of data at least as extreme as observed, assuming the null and model assumptions

Question 6

Which statement about correlation is correct?

A. A correlation of zero proves no relationship of any kind

B. Correlation is unaffected by outliers

C. Correlation measures linear association and does not by itself establish causation

D. Correlation has units equal to the product of the two variables’ units

Correct Answer: C. Correlation measures linear association and does not by itself establish causation

Question 7

In simple linear regression with an intercept, the slope represents:

A. The expected change in the response for a one-unit predictor increase

B. The percentage of observations correctly classified

C. The variance of the response

D. The p-value of the model

Correct Answer: A. The expected change in the response for a one-unit predictor increase

Question 8

Which condition supports the formal ordinary least-squares solution (XᵀX)⁻¹Xᵀy?

A. X has full column rank

B. All predictors are identical

C. The response is always binary

D. The sample size is exactly one

Correct Answer: A. X has full column rank

Question 9

In Bayesian inference, the posterior distribution is proportional to:

A. Prior divided by likelihood

B. Likelihood multiplied by prior

C. Evidence divided by prior

D. Sampling error multiplied by p-value

Correct Answer: B. Likelihood multiplied by prior

Question 10

If a sample has standard deviation 10 and size 100, the estimated standard error of its mean is:

A. 0.1

B. 1

C. 10

D. 100

Correct Answer: B. 1