Chi-Square Homogeneity or Independence (Test) - AP Statistics

$(r-1)(c-1)$, where $r$ and $c$ are the number of rows and columns. Accounts for the number of cells that can vary independently in the table.

All expected counts should be at least 5. Ensures the chi-square approximation is valid for the test statistic.

$E = \frac{(\text{row total})(\text{column total})}{\text{grand total}}$. Based on independence assumption, uses marginal totals to find expected values.

Categorical data. Chi-square tests work only with frequency data from categories.

$\alpha = 0.05$. Standard significance level providing 95% confidence in results.

7.815. Found using chi-square distribution table with 3 degrees of freedom.

Expected count condition. Ensures sufficient sample size for valid chi-square approximation.

Reject the null hypothesis. Large chi-square values provide evidence against independence or homogeneity.

To determine if two categorical variables are independent. Tests whether knowing one variable helps predict the other variable.

$(3-1)(4-1) = 6$. Uses the standard formula for degrees of freedom in contingency tables.

The discrepancy between observed and expected frequencies. Quantifies how much the data deviates from independence assumptions.

If the chi-square statistic > critical value. When test statistic exceeds critical value, evidence contradicts null hypothesis.

The distributions are the same. Assumes no difference exists between population distributions being compared.

$(r-1)(c-1)$. Standard calculation method for contingency table degrees of freedom.

Fail to reject the null hypothesis. Insufficient evidence to conclude variables are dependent or distributions differ.

Independence of observations. Each observation must be independent of all other observations.

To determine the rejection region. Establishes the threshold for deciding statistical significance.

Reject the null hypothesis. Strong evidence against null hypothesis leads to rejection.

To compare them with observed frequencies. Expected frequencies represent what we'd see if null hypothesis were true.

When expected frequencies are less than 5. Low expected frequencies violate the conditions needed for valid results.

$p$-value is compared to $\alpha$ to make a decision. If $p$-value < $\alpha$, reject null; otherwise fail to reject.

Based on the research question. Independence tests one sample; homogeneity compares multiple populations.

Chi-square test for independence. Specifically tests whether two categorical variables are related.

$\alpha = 0.05$. Widely accepted standard for balancing Type I and Type II error risks.

Independence of observations. Each observation must be independent of all other observations.

$\alpha = 0.05$. Widely accepted standard for balancing Type I and Type II error risks.

Chi-square test. Chi-square tests specifically analyze categorical frequency data patterns.

To determine the rejection region. Establishes the threshold for deciding statistical significance.

Testing independence or homogeneity. Primary applications involve testing relationships in categorical data.