Statistics for Two Categorical Variables - AP Statistics

Expected count = $\frac{(\text{row total}) (\text{column total})}{\text{grand total}}$. Assumes independence to calculate what count would be expected in each cell.

A relationship between two categorical variables. Variables are dependent when one variable's distribution changes across levels of another.

To test if there is an association between two categorical variables. Determines whether observed frequencies differ significantly from expected under independence.

All expected cell counts should be at least 5. Ensures the chi-square distribution is a good approximation for the test statistic.

The difference between observed and expected frequencies. Larger values indicate greater deviation from what independence would predict.

$(3-1)(4-1) = 6$. Uses formula $df = (r-1)(c-1)$ where $r$ and $c$ are rows and columns.

The two categorical variables are not independent. Claims there is some relationship or association between the variables.

There is evidence of an association between the variables. The variables are dependent rather than independent of each other.

$\text{Chi-square} = \frac{(O - E)^2}{E}$ summed over all cells. Sums squared standardized deviations across all cells in the table.

Observed counts are the actual data collected in the study. These are the frequencies actually recorded in the research study.

Expected counts are under the assumption of independence. Represent what frequencies would occur if variables were truly independent.

Right-skewed, becoming more symmetric with higher df. Always positive with mean equal to degrees of freedom.

True, a high chi-square indicates strong association. Larger test statistics provide stronger evidence against independence.

A cell is an intersection of a row and a column in the table. Contains the count for one specific combination of the two variable categories.

Expected count = $\frac{(20)(30)}{200} = 3$. Multiplies marginal totals and divides by the overall sample size.

Chi-square statistic. Measures how much observed frequencies deviate from expected frequencies.

The value that separates the region where the null hypothesis is rejected. Determines the boundary for statistical significance in hypothesis testing.

The probability of observing a chi-square statistic as extreme as the one calculated. Used to determine statistical significance by comparing to alpha level.

If p-value < significance level, reject the null hypothesis. Compare p-value to chosen significance level to make decision.

There is a 3% chance the observed association is due to random chance. Low probability suggests the association is unlikely due to chance alone.

Strong evidence against the null hypothesis. Small p-values suggest the null hypothesis is likely false.

To determine if there is enough evidence to reject a null hypothesis. Provides statistical framework for making decisions about population parameters.

Larger sample sizes generally increase the chi-square statistic. More data can detect smaller associations as statistically significant.

Chi-square = $\frac{(15 - 10)^2}{10} = 2.5$. Applies the formula for one cell contribution to the overall statistic.

Rejecting the null hypothesis when it is true. False positive - finding significance when none actually exists.

The distribution of values of one variable among all individuals. Found by summing across rows or columns to get totals for each category.

Failing to reject the null hypothesis when it is false. False negative - missing a real association that actually exists.

Data are a random sample from the population. Ensures results can be generalized to the broader population.

A result that leads to the rejection of the null hypothesis. Indicates sufficient evidence to conclude variables are associated.