Parametric tests are powerful tools for comparing means and variances between groups. T-tests, z-tests, and F-tests help researchers determine if differences are statistically significant, allowing for meaningful conclusions about population parameters based on sample data.
These tests form the backbone of hypothesis testing in statistics. By understanding their applications and assumptions, you'll be equipped to analyze data, make inferences, and draw valid conclusions in various research scenarios.
T-tests
Types of T-tests
- Independent samples t-test compares means between two independent groups (treatment vs. control) to determine if there is a significant difference
- Paired samples t-test compares means between two related groups (pre-test vs. post-test) to assess if a significant change occurred within subjects
- One-sample t-test compares the mean of a single group to a known population mean to evaluate if the sample differs significantly from the population
- Useful when the population standard deviation is unknown and the sample size is small (n < 30)
T-test Considerations
- Degrees of freedom represent the number of independent values that can vary in a statistical calculation
- For independent samples t-test with equal sample sizes, degrees of freedom = $(n_1 + n_2) - 2$, where $n_1$ and $n_2$ are the sample sizes of the two groups
- For paired samples t-test, degrees of freedom = $n - 1$, where $n$ is the number of pairs
- Assumptions of parametric tests must be met for t-tests to be valid
- Independence of observations
- Normality of data distribution within each group
- Homogeneity of variances between groups (equal variances assumed)
- Continuous dependent variable measured on an interval or ratio scale
Z-tests and F-tests
Z-tests
- Z-test compares a sample mean to a known population mean when the population standard deviation is known and the sample size is large (n ≥ 30)
- Assumes a normal distribution of the data
- Useful for testing hypotheses about proportions or means in large samples
- Z-test can also compare two independent proportions to determine if they are significantly different (e.g., success rates between two treatments)
F-tests and ANOVA
- F-test compares the variances of two or more groups to assess if they are significantly different
- Used in the context of Analysis of Variance (ANOVA) to compare means across multiple groups simultaneously
- ANOVA tests the null hypothesis that all group means are equal against the alternative hypothesis that at least one group mean differs
- One-way ANOVA compares means across one independent variable (factor) with three or more levels (groups)
- Two-way ANOVA examines the effects of two independent variables on a dependent variable, including main effects and interaction effects
- Post-hoc tests (Tukey's HSD, Bonferroni correction) are conducted after a significant ANOVA result to determine which specific group means differ from each other while controlling for Type I error rate