ST 231 Winter 2026 Exam Guide: Key Topics in College Statistics

Probability

Basic Probability

Probability is the study of randomness and uncertainty, quantifying the likelihood of events occurring. It forms the foundation for statistical inference and decision-making under uncertainty.

• Probability of an Event (A): The chance that event A occurs, denoted as P(A).

• Sample Space: The set of all possible outcomes of an experiment.

• Complementary Events: The probability that event A does not occur is 1 - P(A).

• Example: The probability of rolling a 4 on a fair six-sided die is 1/6.

Conditional Probability

Conditional probability measures the likelihood of an event given that another event has occurred.

• Formula: $P(A|B) = \frac{P(A \cap B)}{P(B)}$

• Example: If 30% of students are female and 10% are female math majors, the probability a student is a math major given she is female is 10%/30% = 1/3.

Independence

Two events are independent if the occurrence of one does not affect the probability of the other.

• Mathematical Definition: $P(A \cap B) = P(A) \times P(B)$

• Example: Flipping two coins; the result of one does not affect the other.

Discrete Distributions

Discrete probability distributions describe the probabilities of outcomes for discrete random variables.

• Binomial Distribution: Models the number of successes in a fixed number of independent Bernoulli trials.

• Formula: $P(X = k) = {n \choose k} p^k (1-p)^{n-k}$

• Example: Probability of getting 3 heads in 5 coin tosses.

Continuous Distributions

Continuous distributions describe probabilities for continuous random variables, where outcomes can take any value in an interval.

• Normal Distribution: Symmetrical, bell-shaped distribution characterized by mean (μ) and standard deviation (σ).

• Formula: $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} \left(\frac{x-\mu}{\sigma}\right)^2}$

• Student t Distribution: Used when estimating the mean of a normally distributed population with unknown variance and small sample size.

• Example: Heights of adult males are approximately normally distributed.

Box Plots

Box plots are graphical representations of data distributions, showing the median, quartiles, and potential outliers.

• Key Components: Minimum, Q1, Median, Q3, Maximum, and outliers.

• Example: Comparing test scores across different classes.

Confidence Intervals

One Mean with σ Known

Confidence intervals estimate the range in which a population parameter lies, based on sample data.

• Formula: $\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$

• z*: Critical value from the standard normal distribution.

• Example: Estimating average height with known population standard deviation.

One Mean with σ Unknown (Small Sample Size)

• Formula: $\bar{x} \pm t^* \frac{s}{\sqrt{n}}$

• t*: Critical value from the t-distribution with n-1 degrees of freedom.

• Example: Estimating mean exam score from a small class sample.

Two Means

• Formula (Independent Samples): $(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

• Example: Comparing average test scores between two schools.

One Proportion

• Formula: $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Example: Estimating the proportion of students who pass an exam.

Two Proportions

• Formula: $(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

• Example: Comparing pass rates between two different classes.

Hypothesis Testing

One Mean with σ Known

Hypothesis testing evaluates claims about population parameters using sample data.

• Test Statistic: $z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$

• Example: Testing if the average height differs from a known value.

One Mean with σ Unknown (Small Sample Size)

• Test Statistic: $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$

• Example: Testing if a small sample mean differs from a hypothesized value.

Two Means

• Test Statistic (Independent Samples): $t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

• Example: Testing if two groups have different average scores.

One Proportion

• Test Statistic: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

• Example: Testing if the proportion of students passing is different from 50%.

Two Proportions

• Test Statistic: $z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$

• Example: Testing if two classes have different pass rates.

ANOVA (Analysis of Variance)

ANOVA tests for differences among means of three or more groups.

• F-statistic: Ratio of variance between groups to variance within groups.

• Example: Comparing average scores across multiple teaching methods.

Linear Regression

Linear regression models the relationship between a dependent variable and one or more independent variables.

• Simple Linear Regression Equation: $y = \beta_0 + \beta_1 x + \varepsilon$

• Example: Predicting exam scores based on hours studied.

Chi-Square Test (Two-Way Tables for Independence)

The chi-square test assesses whether two categorical variables are independent.

• Test Statistic: $\chi^2 = \sum \frac{(O - E)^2}{E}$

• O: Observed frequency; E: Expected frequency.

• Example: Testing if gender and major are independent in a student population.

Topic	Key Formula	Example Application
Binomial Probability	$P(X = k) = {n \choose k} p^k (1-p)^{n-k}$	Number of heads in coin tosses
Normal Distribution	$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} \left(\frac{x-\mu}{\sigma}\right)^2}$	Heights of adults
Confidence Interval (Mean, σ known)	$\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$	Estimating average height
t-Test (Small Sample)	$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$	Testing mean exam score
ANOVA	F-statistic	Comparing means across groups
Chi-Square Test	$\chi^2 = \sum \frac{(O - E)^2}{E}$	Testing independence in two-way tables

Additional info: Topics such as "Student t distribution," "ANOVA," and "Chi-square test" are expanded with standard academic context to ensure completeness. The table summarizes key formulas and applications for quick review.