Statistics Midterm 2 Review: Probability, Sampling, Regression, and Applications

Probability and Contingency Tables

Analyzing Categorical Data with Contingency Tables

Contingency tables are used to summarize the relationship between two categorical variables. They display the frequency distribution of variables and help in calculating probabilities and testing independence.

• Definition: A contingency table is a matrix that displays the frequency of different combinations of two categorical variables.

• Example: The table below shows AP Statistics students grouped by whether they ate breakfast and their gender.

	Left	Right	Total
Guy	14	6	20
Girl	10	14	24
Total	24	20	44

• Calculating Probabilities: To find the probability that a randomly selected student is a girl who ate breakfast, use $P( ext{Girl} \cap ext{Ate}) = \frac{10}{44}$.

• Independence: Two variables are independent if $P(A \cap B) = P(A)P(B)$.

Probability Rules and Binomial Probability

Basic Probability Concepts

Probability quantifies the likelihood of events occurring. The sum of probabilities for all possible outcomes is 1.

• Key Formula: For independent events, $P(A \cap B) = P(A)P(B)$.

• Binomial Probability: Used when there are a fixed number of independent trials, each with the same probability of success.

Binomial Probability Formula:

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• Example: Probability that at least one of two students did not eat breakfast: $1 - P( ext{both ate breakfast})$.

Sampling Distributions and Central Limit Theorem

Sampling and Standard Deviation

Sampling distributions describe the distribution of a statistic (like the mean) from repeated samples. The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases.

• Standard Deviation of the Sum: For independent random variables $X$ and $Y$, $Var(X+Y) = Var(X) + Var(Y)$, so $SD(X+Y) = \sqrt{SD(X)^2 + SD(Y)^2}$.

• CLT Formula: If $X_1, X_2, ..., X_n$ are independent and identically distributed with mean $\mu$ and standard deviation $\sigma$, then the sample mean $\bar{X}$ has mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt{n}}$.

$\bar{X} \sim N\left(\mu, \frac{\sigma}{\sqrt{n}}\right)$

• Application: Used to estimate probabilities about sample means and sums.

Linear Regression

Regression Analysis and Model Fitting

Linear regression models the relationship between a dependent variable and one or more independent variables. The fitted regression equation predicts the value of the dependent variable.

• Regression Equation: $y = b_0 + b_1x$

• Example: Predicting salary from college GPA: $\text{salary} = 2050 + 1300 \times \text{GPA}$

• Interpretation: The slope $b_1$ represents the change in salary for each unit increase in GPA.

Conditional Probability and Independence

Conditional Probability

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

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

• Independence: Events $A$ and $B$ are independent if $P(A|B) = P(A)$.

• Example: Probability that an accident involved alcohol but not speeding: $P(A \cap \bar{S}) = P(A) - P(A \cap S)$

Expected Value and Variance

Calculating Expected Value and Variance

Expected value is the mean of a random variable's probability distribution. Variance measures the spread.

• Expected Value: $E(X) = \sum x_i P(x_i)$

• Variance: $Var(X) = E[(X - E(X))^2]$

• Example: Expected number of repairs per year: $E(X) = \sum x_i P(x_i)$

Normal Approximation to the Binomial

Using the Normal Model for Binomial Probabilities

When the number of trials is large and $p$ is not too close to 0 or 1, the binomial distribution can be approximated by a normal distribution.

• Normal Approximation: $X \sim N(np, \sqrt{np(1-p)})$

• Application: Used to estimate probabilities for large sample sizes.

Decision Trees and Probability Models

Tree Diagrams for Sequential Events

Tree diagrams help visualize and calculate probabilities for multi-stage events.

• Example: Calculating the probability of success in a strategic decision (e.g., attack vs. not attack) using a tree diagram.

Summary Table: Key Probability Formulas

Concept	Formula	Application
Binomial Probability	$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$	Probability of k successes in n trials
Expected Value	$E(X) = \sum x_i P(x_i)$	Mean of a random variable
Variance (Sum)	$Var(X+Y) = Var(X) + Var(Y)$	Variance of sum of independent variables
Conditional Probability	$P(A|B) = \frac{P(A \cap B)}{P(B)}$	Probability of A given B
Regression Equation	$y = b_0 + b_1x$	Predicting y from x
CLT (Sample Mean)	$\bar{X} \sim N\left(\mu, \frac{\sigma}{\sqrt{n}}\right)$	Distribution of sample mean

Applications and Examples

• Blood Drive: Calculating the probability that at least one of the first 20 donors has Type B blood using binomial and normal approximation.

• Repair Service: Finding expected number of repairs and variance over multiple years.

• Regression: Predicting salary from GPA and evaluating model fit.

Additional info: Some explanations and formulas have been expanded for clarity and completeness. All major topics align with college-level statistics curriculum, including probability, sampling, regression, and applications.