Statistics Essentials Practice Exam

Question 1 Statistics Essentials

Discrete vs continuous random variable?

Correlation of a time series with a lagged version of itself. Bias from differences between people who respond and those who don't. Discrete takes countable values; continuous takes values in a continuous range. Events that cannot occur together; P(A and B) = 0.

Question 2 Statistics Essentials

What is a likelihood ratio test?

p̂ ± z·√(p̂(1−p̂)/n) for large samples. A data collection method using questions posed to respondents. (x − μ) / σ; how many standard deviations a value is from the mean. A hypothesis test comparing likelihoods under two nested models.

Question 3 Statistics Essentials

What is double-blind?

A numerical value that describes a characteristic of a sample (e.g., x̄, s). Both participants and researchers don't know who is in which group. A variable whose value is a numerical outcome of a random phenomenon. To prevent positive and negative deviations from cancelling out and to weight larger deviations more heavily.

Question 4 Statistics Essentials

What is overfitting?

Failing to reject H₀ when it is false (false negative). The average of squared deviations from the mean; measures spread. The distribution of a single variable obtained by summing across the other in a contingency table. A model that captures noise in training data and generalizes poorly.

Question 5 Statistics Essentials

What is publication bias?

A variable whose values are categories (e.g., color, gender). A systematic error from non-random selection that distorts estimates. The tendency for journals to publish only statistically significant or positive findings. Regression with two or more independent variables.

Question 6 Statistics Essentials

What is the F-distribution?

A measure of the influence of a point on regression coefficients. Bias from differences between people who respond and those who don't. The tendency for journals to publish only statistically significant or positive findings. The distribution of the ratio of two scaled chi-square variables; used in ANOVA and regression.

Question 7 Statistics Essentials

What is Akaike Information Criterion (AIC)?

Mean = np; variance = np(1−p). A model-selection metric: 2k − 2 ln(L); lower is better, penalizing complexity. E[X] = Σ x · P(X = x) for discrete variables; the long-run average. The sampling distribution of the mean approaches a normal distribution as sample size increases, regardless of the population's shape.

Question 8 Statistics Essentials

What is the Bonferroni correction?

P(data | hypothesis); how plausible the data are under a hypothesis. Dividing α by the number of tests to control family-wise error. Selecting items without putting them back; changes probabilities across selections. A smoothed series computed by averaging values within a window.

Question 9 Statistics Essentials

What is a Type II error?

The probability of a hypothesis before seeing the data. A sample where every member of the population has an equal chance of being chosen. Failing to reject H₀ when it is false (false negative). E[X] = Σ x · P(X = x) for discrete variables; the long-run average.

Question 10 Statistics Essentials

What is cluster sampling?

Randomly selecting whole groups (clusters) and surveying everyone in them; reduces cost. The number of values free to vary when estimating a statistic; often n − 1 for variance estimates. Bias from differences between people who respond and those who don't. The entire set of individuals, items, or measurements that share a common property and are the subject of a statistical study.

Question 11 Statistics Essentials

What is a contingency table?

log(p / (1 − p)); the link function in logistic regression. A function that gives the probability of each possible value of a discrete random variable. A table cross-tabulating frequencies of two categorical variables. A method of estimating model performance by splitting data into training and validation folds.

Question 12 Statistics Essentials

What is the standard normal distribution?

A normal distribution with mean 0 and standard deviation 1. A state in a Markov chain that, once entered, cannot be left. A variable measured on a numeric scale (e.g., height, age). Bias from differences between people who respond and those who don't.

Question 13 Statistics Essentials

What is statistical power?

Bias from differences between people who respond and those who don't. A measure of the influence of a point on regression coefficients. The expected change in y for a one-unit change in that x, holding other variables constant. The probability of correctly rejecting a false null (1 − β).

Question 14 Statistics Essentials

What is a parameter?

A numerical value that describes a characteristic of a population (e.g., μ for mean, σ for standard deviation). Correlation of a time series with a lagged version of itself. A variable whose value is a numerical outcome of a random phenomenon. The sampling distribution of the mean approaches a normal distribution as sample size increases, regardless of the population's shape.

Question 15 Statistics Essentials

What is a Bayesian interpretation of probability?

A design where different subjects are assigned to different conditions. Probability as a degree of belief, updated with evidence. A hypothesis test comparing likelihoods under two nested models. Bessel's correction; it makes the sample variance an unbiased estimator of population variance.

Question 16 Statistics Essentials

What is the range?

The difference between the maximum and minimum values in a dataset. A graphical display of effect estimates and confidence intervals from multiple studies in a meta-analysis. A model that captures noise in training data and generalizes poorly. When population σ is known or n is large.

Question 17 Statistics Essentials

What is a Markov chain?

A stochastic process where the next state depends only on the current state. Finding structure in unlabeled data (e.g., clustering, PCA). A symmetric distribution similar to the normal but with heavier tails; used when sample size is small or σ is unknown. A test that does not assume a specific distribution (e.g., normality) for the data.

Question 18 Statistics Essentials

What is conditional probability?

(x − μ) / σ; how many standard deviations a value is from the mean. The probability of A given B: P(A|B) = P(A and B) / P(B). P(A) = Σ P(A|B_i)·P(B_i) over a partition {B_i}. E[X] = Σ x · P(X = x) for discrete variables; the long-run average.

Question 19 Statistics Essentials

Define the median.

The middle value of an ordered dataset; if even count, the average of the two middle values. Selecting items without putting them back; changes probabilities across selections. An observation with an extreme x-value in regression. Methods for summarizing and describing the main features of a dataset.

Question 20 Statistics Essentials

Define the mean.

A plot of pairs of numerical values to visualize the relationship between two variables. The arithmetic average: the sum of values divided by the number of values. Regression with two or more independent variables. A model for the count of events occurring in a fixed interval at a constant rate λ.

Question 21 Statistics Essentials

What is a Bayesian network?

A design where different subjects are assigned to different conditions. No linear association (variables may still have a non-linear relationship). A graphical model encoding conditional dependencies among variables. (x − μ) / σ; how many standard deviations a value is from the mean.

Question 22 Statistics Essentials

Why divide by n − 1 in sample variance?

Conducting many tests, which inflates the chance of a Type I error. Bessel's correction; it makes the sample variance an unbiased estimator of population variance. An observation far from the bulk of the data. The probability of A given B: P(A|B) = P(A and B) / P(B).

Question 23 Statistics Essentials

What is the formula for the standard error of a proportion?

√(p(1−p)/n). The proportion of variance in the dependent variable explained by the model. A single value used as an estimate of a parameter (e.g., sample mean). A symmetric distribution similar to the normal but with heavier tails; used when sample size is small or σ is unknown.

Question 24 Statistics Essentials

What is the addition rule?

A hypothesis test comparing likelihoods under two nested models. The distribution of a single variable obtained by summing across the other in a contingency table. P(A or B) = P(A) + P(B) − P(A and B). Mean = np; variance = np(1−p).

Question 25 Statistics Essentials

What is autocorrelation?

A model-selection metric: k·ln(n) − 2 ln(L); penalizes complexity more heavily than AIC. The square root of the variance; spread expressed in the original units. exp(β); the multiplicative change in odds for a one-unit increase in the predictor. Correlation of a time series with a lagged version of itself.

Question 26 Statistics Essentials

What is regularization?

Adding a penalty for complexity to a model to discourage overfitting (e.g., L1, L2). The entire set of individuals, items, or measurements that share a common property and are the subject of a statistical study. A balance between error from overly simple models (bias) and overly complex models (variance). A range of plausible values for a parameter, with an associated confidence level.

Question 27 Statistics Essentials

What is the bias-variance tradeoff?

An observation with an extreme x-value in regression. A balance between error from overly simple models (bias) and overly complex models (variance). A statistical synthesis of results from multiple studies on the same question. Finding structure in unlabeled data (e.g., clustering, PCA).

Question 28 Statistics Essentials

What is bias of an estimator?

The expected difference between the estimator and the true parameter. (x − μ) / σ; how many standard deviations a value is from the mean. The omitted category against which other dummy levels are compared. A state in a Markov chain that, once entered, cannot be left.

Question 29 Statistics Essentials

What is systematic sampling?

A plot of residuals vs fitted values; used to check assumptions of regression. No linear association (variables may still have a non-linear relationship). A third variable that influences both the independent and dependent variables, creating a spurious association. Selecting every kth element after a random start.

Question 30 Statistics Essentials

What is multiple testing?

A value below which a given percentage of observations fall. A trend in aggregated data that reverses when subgroups are considered. Conducting many tests, which inflates the chance of a Type I error. AutoRegressive Integrated Moving Average; a class of time-series models combining AR and MA components with differencing.

Question 31 Statistics Essentials

What is the Wilcoxon signed-rank test?

The evidence is strong enough (p < α) to favor H₁; not proof, but strong evidence. A non-parametric paired-sample test based on ranks. The probability of observing data at least as extreme as the sample, assuming H₀ is true. An observation with an extreme x-value in regression.

Question 32 Statistics Essentials

What does the confidence level affect?

A table cross-tabulating frequencies of two categorical variables. Adding a penalty for complexity to a model to discourage overfitting (e.g., L1, L2). The width of the interval: higher confidence → wider interval. A measure of the strength and direction of linear association between two variables.

Question 33 Statistics Essentials

What are post-hoc tests?

−1 to +1. The ratio of between-group variance to within-group variance. Pairwise comparisons (e.g., Tukey's HSD) used after a significant ANOVA. A numerical value that describes a characteristic of a sample (e.g., x̄, s).

Question 34 Statistics Essentials

What is supervised learning?

Bias from differences between people who respond and those who don't. A function whose area over an interval gives the probability of a continuous random variable falling in that interval. The width of the interval: higher confidence → wider interval. Learning a function from labeled input-output pairs.

Question 35 Statistics Essentials

What is the Kruskal-Wallis test?

An observation with an extreme x-value in regression. A non-parametric alternative to one-way ANOVA. E[X] = Σ x · P(X = x) for discrete variables; the long-run average. Manipulating data analysis until non-significant results become significant.

Question 36 Statistics Essentials

What does "reject H₀" mean?

P(not A) = 1 − P(A). A subset of a population selected to represent it for analysis. A non-parametric test that builds the null distribution by shuffling labels. The evidence is strong enough (p < α) to favor H₁; not proof, but strong evidence.

Question 37 Statistics Essentials

What is a histogram?

(x − μ) / σ; how many standard deviations a value is from the mean. The sampling distribution of the mean approaches a normal distribution as sample size increases, regardless of the population's shape. √(p(1−p)/n). A bar chart of a frequency distribution for a continuous variable, with bars touching.

Question 38 Statistics Essentials

What is k-means clustering?

E[X] = Σ x · P(X = x) for discrete variables; the long-run average. The expected difference between the estimator and the true parameter. An algorithm partitioning data into k clusters by minimizing within-cluster variance. A balance between error from overly simple models (bias) and overly complex models (variance).

Question 39 Statistics Essentials

Define the mode.

The value or values that occur most frequently in a dataset. A plot used to detect publication bias in meta-analyses. A data collection method using questions posed to respondents. A theoretical lower bound on the variance of any unbiased estimator.

Question 40 Statistics Essentials

What is an interaction term in regression?

A product term (x1·x2) capturing how one predictor's effect depends on another. A non-parametric test that builds the null distribution by shuffling labels. A procedure for using data to evaluate whether a claim about a population is plausible. The sampling distribution is the distribution of a statistic over many samples; the population distribution describes the underlying data.

Question 41 Statistics Essentials

What is a positively skewed distribution?

A test of whether two categorical variables are independent. Hypothesizing After Results are Known; presenting post-hoc findings as a priori. A distribution with a long right tail; mean > median. log(p / (1 − p)); the link function in logistic regression.

Question 42 Statistics Essentials

What is a statistic (in this context)?

A numerical value that describes a characteristic of a sample (e.g., x̄, s). Ignoring base-rate information in favor of specific-case information. P(A and B) = P(A)·P(B); knowing B tells you nothing about A. A design where different subjects are assigned to different conditions.

Question 43 Statistics Essentials

What is a random variable?

A variable whose value is a numerical outcome of a random phenomenon. Adds |β| penalty; can shrink coefficients to zero, doing feature selection. Yes; E[x̄] = μ. The square root of the variance; spread expressed in the original units.

Question 44 Statistics Essentials

What is stationarity?

A field combining statistics and machine learning for prediction and inference. A subset of a population selected to represent it for analysis. A table cross-tabulating frequencies of two categorical variables. A time series whose statistical properties (mean, variance) don't change over time.

Question 45 Statistics Essentials

Why do we use samples instead of populations?

Because measuring the entire population is often impractical, costly, or impossible. The distribution of the sum of squared standard normal variables; used in tests of variance and categorical data. Bessel's correction; it makes the sample variance an unbiased estimator of population variance. The mistaken belief that past random events affect future independent random events.

Question 46 Statistics Essentials

What is the law of small numbers?

A balance between error from overly simple models (bias) and overly complex models (variance). An experimental design where each subject experiences all conditions. A cognitive bias: assuming small samples reflect population properties. Methods for summarizing and describing the main features of a dataset.

Question 47 Statistics Essentials

Mean and variance of Poisson(λ)?

An algorithm partitioning data into k clusters by minimizing within-cluster variance. A measure of the influence of a point on regression coefficients. Both equal to λ. A bias from inaccurate or untruthful responses, e.g., social desirability.

Question 48 Statistics Essentials

What is the complement rule?

P(not A) = 1 − P(A). Dividing α by the number of tests to control family-wise error. The claim we are testing for evidence, denoted H₁ or Ha. The omitted category against which other dummy levels are compared.

Question 49 Statistics Essentials

What is a confounding variable?

A third variable that influences both the independent and dependent variables, creating a spurious association. If we repeated sampling many times, about 95% of such intervals would contain the true parameter. 80% of the variability in the response is explained by the model. A function whose area over an interval gives the probability of a continuous random variable falling in that interval.

Question 50 Statistics Essentials

What is ARIMA?

A bias from inaccurate or untruthful responses, e.g., social desirability. P(A) = Σ P(A|B_i)·P(B_i) over a partition {B_i}. A variable whose value is a numerical outcome of a random phenomenon. AutoRegressive Integrated Moving Average; a class of time-series models combining AR and MA components with differencing.