STAT 118: Exam 3 Study Guide (Chapters 3, 12-16)

Chapter 3: Relationships Between Categorical Variables – Contingency Tables

Contingency Tables

Contingency tables are a fundamental tool for analyzing the relationship between two categorical variables. They display the frequency distribution of variables and allow for the calculation of row, column, and table percentages.

• Definition: A contingency table is a matrix that shows the frequency of occurrence of combinations of categorical variables.

• Key Uses:

  • Obtaining row, column, and table percents

  • Identifying associations between variables

  • Spotting phenomena such as Simpson's paradox

• Visualization: Stacked bar plots are often used to visualize contingency tables.

• Example: A table showing the number of males and females who prefer different types of music genres.

Chapter 12: From Randomness to Probability / Linear Model

Randomness and Probability

Probability theory provides a mathematical framework for quantifying uncertainty and randomness in experiments and phenomena.

• Random Phenomenon: An event whose outcome cannot be predicted with certainty.

• Key Terms:

  • Trial: A single attempt or realization of a random phenomenon.

  • Outcome: The result of a trial.

  • Sample Space: The set of all possible outcomes.

  • Event: A subset of the sample space.

  • Law of Large Numbers: As the number of trials increases, the relative frequency of an event approaches its probability.

• Probability Rules:

  • Probability of an event is between 0 and 1: $0 \leq P(A) \leq 1$

  • Probability Assignment Rule: $P(S) = 1$ (where S is the sample space)

  • Addition Rule (for mutually exclusive events): $P(A \text{ or } B) = P(A) + P(B)$

  • General Addition Rule: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

• Conditional Probability: $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$

• Joint Probability: $P(A \text{ and } B) = P(A|B) \times P(B)$

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

• Multiplication Rule: If A and B are independent, $P(A \text{ and } B) = P(A) \times P(B)$

Chapter 13: Sampling Distribution Models & Confidence Intervals for Proportions

Sampling Variability and Distribution

Sampling distributions describe the variability of sample statistics from repeated samples of the same population.

• Sampling Distribution for a Proportion: If conditions are met, the sampling distribution of the sample proportion $\hat{p}$ is approximately normal: $\hat{p} \sim N\left(p, \sqrt{\frac{p(1-p)}{n}}\right)$

• Conditions:

  • Randomization condition

  • Success/failure condition: $np \geq 10$ and $n(1-p) \geq 10$

• Standard Error of $\hat{p}$: $SD(\hat{p}) = \sqrt{\frac{p(1-p)}{n}} = SE(\hat{p})$

Confidence Interval for the Population Proportion $p$

A confidence interval estimates the range in which the true population proportion is likely to fall.

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

• Finding the critical value ($z^*$): Use $z^*$ for 95% CI (commonly $z^* = 1.96$)

• Interpretation: The confidence interval gives a range of plausible values for the population proportion.

Chapter 14: Confidence Intervals for Means

Central Limit Theorem and Sampling Distribution for a Mean

The Central Limit Theorem states that the sampling distribution of the sample mean approaches normality as sample size increases, regardless of the population's distribution.

• Sampling Distribution for a Mean: $\bar{x} \sim N\left(\mu, \frac{\sigma}{\sqrt{n}}\right)$

• Conditions:

  • Randomization condition

  • Nearly Normal condition

If the population distribution is...	Then the condition is generally met for...
Fairly symmetric and no clear outliers	All n, including n < 15
Skewed and no clear outliers	n > 30
Extremely skewed and clear outliers	n > 40

• Standard Error of $\bar{x}$: $SD(\bar{x}) = \frac{\sigma}{\sqrt{n}} = SE(\bar{x})$

Student's t Distribution

When the population standard deviation is unknown, the t distribution is used for inference about means.

• Formula for t-statistic: $t = \frac{\bar{x} - \mu}{SE(\bar{x})}$

• Degrees of Freedom: $df = n - 1$

• Finding the critical value ($t^*$): Use t-tables or software for the desired confidence level.

Chapter 15: Testing Hypotheses

Hypothesis Test Logic

Hypothesis testing is a formal procedure for comparing observed data to a claim about a population parameter.

• Steps to Perform a Hypothesis Test:

  1. State the null hypothesis ($H_0$), alternative hypothesis ($H_A$), and significance level ($\alpha$).

  2. Select the test and check if the conditions are met.

  3. Calculate the test statistic and p-value.

  4. Compare the p-value to $\alpha$ and state your conclusion in context.

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

• Test Statistic for One Sample Test for a Mean: $t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$

If the sample distribution is...	Then the condition is generally met for...
Fairly symmetric and no clear outliers	All n, including n < 15
Skewed and no clear outliers	n > 30
Extremely skewed and clear outliers	n > 40

Chapter 16: Testing Hypotheses (continued)

P-values and Types of Errors

P-values quantify the strength of evidence against the null hypothesis. Understanding errors is crucial for interpreting test results.

• P-value: The probability of observing a test statistic as extreme as, or more extreme than, the observed value under $H_0$.

• Types of Errors:

  • Type I error: Rejecting $H_0$ when it is true (false positive).

  • Type II error: Failing to reject $H_0$ when it is false (false negative).

• Power: The probability of correctly rejecting $H_0$ when it is false.

• Relationship: Power increases with sample size and decreases with lower significance level ($\alpha$).