Glossary
Alternative Hypothesis (Hₐ)
A statement that contradicts the null hypothesis, proposing that there is an effect, a difference, or a relationship between variables.
Example:
The alternative hypothesis for a homogeneity test might state that the distribution of preferred study methods is not the same for high school and college students.
Categorical Variables
Variables that place individuals into categories, rather than measuring numerical quantities.
Example:
Hair color (blonde, brown, black) or type of pet (dog, cat, fish) are examples of categorical variables.
Chi-Squared Test for Homogeneity
A chi-squared test used to compare the distribution of a single categorical variable across two or more independent populations or groups.
Example:
A researcher might use a chi-squared test for homogeneity to determine if the proportion of students who plan to attend college is the same across different high schools in a district.
Chi-Squared Test for Independence
A chi-squared test used to determine if there is a statistically significant association between two categorical variables within a single population.
Example:
To see if there's a relationship between a person's favorite genre of music and their preferred social media platform, you would use a chi-squared test for independence.
Chi-squared (χ²) tests
Statistical tests used to analyze categorical data, often to determine if there's a significant association between variables or if observed frequencies differ from expected frequencies.
Example:
An AP Stats student might use a chi-squared test to see if there's a relationship between a student's favorite subject and their preferred learning style.
Degrees of Freedom (df)
A value that indicates the number of independent pieces of information used to calculate a statistic, crucial for determining the shape of the chi-squared distribution.
Example:
For a 3x4 two-way table, the degrees of freedom would be (3-1)*(4-1) = 6.
Expected Counts
The frequencies that would be anticipated in each cell of a two-way table if the null hypothesis (of no association or homogeneity) were true.
Example:
If 50% of students prefer chocolate ice cream, and you have 100 students, the expected count for chocolate lovers would be 50.
Null Hypothesis (H₀)
A statement of no effect, no difference, or no relationship between variables, which is assumed to be true until evidence suggests otherwise.
Example:
For a chi-squared test of independence, the null hypothesis would state that there is no association between a person's favorite color and their birth month.
Observed Counts
The actual frequencies or counts recorded in each cell of a two-way table from the collected data.
Example:
In a survey of 100 students, if 60 actually said they prefer chocolate ice cream, then 60 is the observed count for that category.
P-value
The probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.
Example:
If a p-value is 0.02, it means there's a 2% chance of seeing your results if the null hypothesis were actually true, leading you to consider rejecting it.
Two-Way Table
A table that displays the counts of observations for two categorical variables, organizing data into rows and columns to show their relationship.
Example:
A two-way table could show the number of students who prefer online vs. in-person classes, broken down by grade level (freshman, sophomore, etc.).