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:

• 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:

• 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:

• 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:

• 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:

• 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:

• 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):

• Example: Comparing average test scores between two schools.

One Proportion

• Formula:

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

Two Proportions

• Formula:

• 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:

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

One Mean with σ Unknown (Small Sample Size)

• Test Statistic:

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

Two Means

• Test Statistic (Independent Samples):

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

One Proportion

• Test Statistic:

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

Two Proportions

• Test Statistic:

• 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:

• 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:

• O: Observed frequency; E: Expected frequency.

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

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.