Introduction to Probability and Statistics

Prerequisites

Content

• Population vs. sample; parameter vs. statistic (\(\mu\) vs. \(\bar{x}\), \(\sigma\) vs. \(s\))
• Qualitative (nominal/ordinal) vs. quantitative (discrete/continuous) variables
• Sampling methods: simple random, stratified, systematic, cluster, multistage, convenience
• Bias in sampling and its effect on validity of conclusions
• Survey design principles

Prerequisites

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• Frequency distributions: absolute, relative (\(\frac{f}{n}\)), and cumulative frequencies
• Histograms (quantitative data) vs. bar charts (qualitative data)
• Pie charts, stem-and-leaf plots, scatter plots, and time series plots
• Graph misrepresentation: truncated axes, inconsistent scales, misleading displays

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• Mean \(\bar{x} = \frac{\sum x_i}{n}\): calculation, properties, sensitivity to outliers
• Median: calculation procedure (sort first, then locate), use with skewed data
• Mode: unimodal, bimodal, multimodal; mode for qualitative data
• Relationship between mean, median, and mode as a function of skewness

Prerequisites

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• Range as a basic but unstable measure of spread
• Population variance \(\sigma^2\) vs. sample variance \(s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}\) — the \(n-1\) denominator
• Population SD \(\sigma\) vs. sample SD \(s\)
• Quartiles (Q1, Q2, Q3), IQR \(=\) Q3 \(-\) Q1, and the five-number summary
• Coefficient of variation; identifying outliers using \(1.5 \times \text{IQR}\) fences

Prerequisites

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• Percentiles, quartiles, and deciles — locating and interpreting
• Z-score (standardization): \(z = \frac{x - \mu}{\sigma}\); interpretation in standard deviations from the mean
• Skewness: right-skewed (tail extends right), left-skewed, symmetric distributions
• Empirical Rule (68–95–99.7) for approximately normal distributions

Prerequisites

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• Sample space and events; classical, empirical, and subjective probability
• Probability rules: \(0 \leq P(A) \leq 1\); \(P(S) = 1\); Addition Rule; Complement Rule: \(P(A') = 1 - P(A)\)
• Compound events: union \(A \cup B\) and intersection \(A \cap B\)
• Mutually exclusive events; Venn diagrams as a reasoning tool

Prerequisites

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• Conditional probability: \(P(A \mid B) = \frac{P(A \cap B)}{P(B)}\)
• General multiplication rule: \(P(A \cap B) = P(B) \cdot P(A \mid B)\)
• Independent events: definition \(P(A \mid B) = P(A)\) and test \(P(A \cap B) = P(A) \cdot P(B)\)
• Tree diagrams for multi-stage experiments

Prerequisites

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• Fundamental Counting Principle for sequential, independent choices
• Factorials: \(n! = n \cdot (n-1) \cdots 1\); convention \(0! = 1\)
• Permutations (order matters): \(_nP_r = \frac{n!}{(n-r)!}\)
• Combinations (order irrelevant): \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\)
• Distinguishing permutations from combinations in context
• Applications to computing probabilities over equally-likely sample spaces

Prerequisites

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• Random variables: discrete vs. continuous; notation \(X\), \(P(X = x)\)
• Probability mass function (PMF) and its properties (\(\sum P(X = x) = 1\))
• Expected value: \(E(X) = \mu = \sum x \cdot P(X = x)\)
• Variance: \(\text{Var}(X) = E(X^2) - [E(X)]^2\); standard deviation \(\sigma_X\)
• Cumulative distribution function (CDF)

Prerequisites

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• Four BINS conditions: Binary outcomes, Independent trials, fixed Number, constant Success probability
• Binomial formula: \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\)
• Mean: \(\mu = np\); variance: \(\sigma^2 = np(1-p)\); SD: \(\sigma = \sqrt{np(1-p)}\)
• Cumulative probabilities: "at most," "at least," "more than," "fewer than"

Prerequisites

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• Properties of the normal distribution: bell-shaped, symmetric about \(\mu\), asymptotic tails
• Standard normal \(Z \sim N(0, 1)\); z-table gives left-tail area \(P(Z < a)\)
• Finding probabilities: \(P(Z < a)\), \(P(Z > a) = 1 - P(Z < a)\), \(P(a < Z < b)\)
• Standardizing: \(z = \frac{x - \mu}{\sigma}\) to transform \(X \sim N(\mu, \sigma^2)\) to \(Z\)
• Finding values given a probability (inverse normal / quantile lookup)

Prerequisites

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• Sampling distribution of \(\bar{x}\): mean \(\mu_{\bar{x}} = \mu\) and standard error \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\)
• Central Limit Theorem: for large \(n\), \(\bar{x}\) is approximately normal regardless of population shape
• Effect of sample size on the shape and spread of the sampling distribution
• Finite population correction (overview)
• Computing probabilities for sample means: \(z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}}\)

Prerequisites

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• Point estimate vs. interval estimate; margin of error \(E = z^* \cdot \frac{\sigma}{\sqrt{n}}\)
• CI formula: \(\bar{x} \pm z^* \cdot \frac{\sigma}{\sqrt{n}}\) (use \(s\) when \(\sigma\) unknown and \(n \geq 30\))
• Critical z-values: \(z^* = 1.645\) (90%), \(z^* = 1.96\) (95%), \(z^* = 2.576\) (99%)
• Correct interpretation: "We are 95% confident that \(\mu\) falls between [L, U]"
• Sample size determination: \(n = \left(\frac{z^* \cdot \sigma}{E}\right)^2\)

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• t-distribution: bell-shaped, heavier tails than normal, indexed by degrees of freedom \(df = n - 1\)
• Critical t-values from t-table; t approaches z as df increases
• CI formula: \(\bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}}\)
• Conditions: approximately normal population, \(\sigma\) unknown
• When to use t vs. z: the practical decision rule

Prerequisites

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• Point estimate: \(\hat{p} = \frac{x}{n}\); standard error \(\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)
• CI formula: \(\hat{p} \pm z^* \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)
• Conditions: \(n\hat{p} \geq 5\) and \(n(1-\hat{p}) \geq 5\)
• Sample size determination using \(p^* = 0.5\) when no prior estimate is available
• One-sided vs. two-sided CIs for proportions

Prerequisites

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• Five steps: state H₀ and Hₐ, check conditions, compute test statistic, find p-value, state conclusion in context
• Test statistic: \(z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}\)
• Type I error (significance level \(\alpha\)) and Type II error (\(\beta\)); the trade-off between them
• p-value: \(P(\text{data this extreme} \mid H_0 \text{ true})\); comparing to \(\alpha\)
• One-tailed vs. two-tailed tests; choosing Hₐ before seeing data

Prerequisites

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• t-test for small sample mean: \(t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}\), \(df = n-1\), five steps, conditions
• Proportion test: \(z = \frac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}}\) — using \(p_0\), not \(\hat{p}\), in the denominator
• One-tailed vs. two-tailed tests for both scenarios
• Approximate p-values from the t-table; exact values with technology

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• Scatter plots: direction, form (linear vs. non-linear), strength, outliers
• Pearson \(r\): range \(-1 \leq r \leq 1\), sign (direction), magnitude (strength)
• Coefficient of determination \(r^2\): proportion of variance in \(y\) explained by the linear relationship with \(x\)
• Spearman's rank correlation (overview for non-normal data and ordinal variables)
• Correlation vs. causation: confounding variables and spurious correlations

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• Least-squares regression line: \(\hat{y} = a + bx\); minimizing the sum of squared residuals
• Slope: \(b = r \cdot \frac{s_y}{s_x}\); intercept: \(a = \bar{y} - b\bar{x}\)
• Residuals: \(e_i = y_i - \hat{y}_i\); sum of squared errors (SSE)
• The regression line always passes through \((\bar{x}, \bar{y})\)
• Conditions and assumptions for simple linear regression

Prerequisites

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• Interpreting slope: a one-unit increase in \(x\) is associated with \(b\) units of change in \(\hat{y}\), on average
• Interpreting intercept: predicted \(y\) when \(x = 0\) (only meaningful when \(x = 0\) is in context)
• Making predictions; interpolation (safe) vs. extrapolation (risky)
• Residual plots for checking linearity and homoscedasticity assumptions
• Outliers and influential points; their effect on slope and intercept
• Significance test for \(r\): \(t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}\), \(df = n-2\); H₀: \(\rho = 0\) (no linear relationship in the population)

Prerequisites

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• Contingency tables (two-way tables): observed frequencies and marginal totals
• Expected frequencies: \(E = \frac{\text{row total} \times \text{column total}}{\text{grand total}}\)
• Chi-square test statistic: \(\chi^2 = \sum \frac{(O - E)^2}{E}\)
• Degrees of freedom: \(df = (r-1)(c-1)\); p-values from the \(\chi^2\) distribution
• Five steps of the chi-square test; condition: all expected frequencies \(\geq 5\)
• Cramér's V as an effect size measure

Prerequisites

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• Decision rule: two quantitative variables → correlation/regression; two qualitative variables → chi-square
• Practical significance vs. statistical significance: effect size matters alongside p-value
• Data collection considerations for bivariate studies
• Interpreting results in real-world context; ethical considerations in analysis
• Presenting and communicating statistical findings responsibly