Hypothesis Testing

Types of Errors in Hypothesis Testing

In hypothesis testing, two types of errors can occur when making decisions about the null hypothesis (H0):

• Type I Error (α): Rejecting the null hypothesis when it is actually true.

• Type II Error (β): Failing to reject the null hypothesis when it is actually false.

Example: If a test incorrectly concludes that a new drug is effective when it is not, this is a Type I error.

Formulating Hypotheses

Hypothesis testing involves two competing statements:

• Null Hypothesis (H0): The statement being tested, usually a statement of no effect or status quo.

• Alternative Hypothesis (Ha): The statement we want to find evidence for.

Example: Testing if the mean weight is greater than 50 kg:

• H0: μ = 50

• Ha: μ > 50

Decisions in Hypothesis Testing

There are two possible decisions:

• Reject H0: Sufficient evidence exists to support Ha.

• Fail to Reject H0: Insufficient evidence to support Ha.

Significance Level (α)

The significance level (α) is the probability of making a Type I error. Common values are 0.05 or 0.01. Lowering α decreases the probability of a Type I error but increases the probability of a Type II error.

P-Value Interpretation

The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value, assuming H0 is true. If the p-value is less than α, reject H0.

Critical Value Approach

Compare the test statistic to a critical value from the appropriate distribution (z or t). If the test statistic falls in the rejection region, reject H0.

Test Statistic Formulas

• z-test (known σ):

• t-test (unknown σ):

Example Table: Hypothesis Test Summary

Confidence Intervals

Definition and Interpretation

A confidence interval provides a range of plausible values for a population parameter. A 95% confidence interval means that, in repeated sampling, 95% of such intervals will contain the true parameter.

Formulas

• Known σ:

• Unknown σ:

Margin of Error (E)

The margin of error is the maximum expected difference between the true population parameter and a sample estimate.

• (for known σ)

• (for unknown σ)

Sample Size Determination

To achieve a desired margin of error, solve for n:

Example: To estimate a mean with σ = 10, E = 2, and 95% confidence (z = 1.96):

• (round up to 97)

Probability Rules

Multiplication Rule

• Independent Events:

• Dependent Events:

Addition Rule

• Mutually Exclusive:

• Not Mutually Exclusive:

Discrete Probability Distributions

Binomial Distribution

• Mean:

• Variance:

Where n = number of trials, p = probability of success, q = 1 - p.

Normal Distribution and Standard Scores

Standard Normal Distribution

The standard normal distribution is a normal distribution with mean 0 and standard deviation 1. Z-scores measure how many standard deviations a value is from the mean:

Use the Standard Normal Table to find probabilities associated with z-scores.

Student's t-Distribution

Used when the population standard deviation is unknown and the sample size is small (n < 30). The t-distribution is wider than the normal distribution and depends on degrees of freedom (df = n - 1).

Correlation and Regression

Correlation Coefficient (r)

Measures the strength and direction of a linear relationship between two variables:

Regression Line

The regression line predicts the value of a dependent variable based on the value of an independent variable:

Statistical Tables

Standard Normal Table

Provides cumulative probabilities for z-scores. Used to find probabilities and critical values for hypothesis testing and confidence intervals.

t-Distribution Table

Gives critical values for the t-distribution based on degrees of freedom and significance level.

Summary Table: Key Formulas

Applications and Examples

• Calculating minimum sample size for a desired margin of error.

• Constructing and interpreting confidence intervals for means.

• Performing hypothesis tests for population means using z or t statistics.

• Using statistical tables to find critical values and probabilities.

Additional info: These notes synthesize content from a practice final exam, formula sheets, and statistical tables, covering hypothesis testing, confidence intervals, probability rules, and applications relevant to a college-level statistics course.