Key Concepts in Statistical Hypothesis Tests

Statistical hypothesis tests help us determine if observed data significantly differs from what we expect. These tests, like Z-tests and T-tests, are essential tools in probability and statistics for making informed decisions based on data analysis.

1. Z-test

   • Used to determine if there is a significant difference between sample and population means when the population variance is known.
   • Applicable for large sample sizes (n > 30) or when the population is normally distributed.
   • Assumes that the data is continuous and follows a normal distribution.

2. T-test (one-sample, two-sample, paired)

   • One-sample T-test: Compares the mean of a single sample to a known population mean.
   • Two-sample T-test: Compares the means of two independent samples to see if they are significantly different.
   • Paired T-test: Compares means from the same group at different times (e.g., before and after treatment).

3. Chi-square test

   • Tests the association between categorical variables by comparing observed frequencies to expected frequencies.
   • Commonly used in contingency tables to assess independence.
   • Requires a minimum sample size and expected frequency in each category.

4. F-test

   • Used to compare the variances of two populations to determine if they are significantly different.
   • Often used in the context of ANOVA to test the equality of variances.
   • Assumes that the data is normally distributed and independent.

5. ANOVA (Analysis of Variance)

   • Compares means across three or more groups to determine if at least one group mean is different.
   • Can be one-way (one independent variable) or two-way (two independent variables).
   • Assumes normality, independence, and homogeneity of variances.

6. Regression analysis

   • Examines the relationship between a dependent variable and one or more independent variables.
   • Can be simple (one independent variable) or multiple (more than one independent variable).
   • Helps in predicting outcomes and understanding relationships between variables.

7. Wilcoxon rank-sum test

   • A non-parametric test that compares the ranks of two independent samples to assess whether their population distributions differ.
   • Used when the assumptions of the T-test are not met (e.g., non-normal data).
   • Suitable for ordinal data or continuous data that do not meet normality assumptions.

8. Kruskal-Wallis test

   • A non-parametric alternative to ANOVA for comparing three or more independent groups.
   • Assesses whether the samples originate from the same distribution.
   • Useful when the assumptions of ANOVA are violated, such as non-normality.

9. Shapiro-Wilk test

   • Tests the null hypothesis that a sample comes from a normally distributed population.
   • Commonly used to assess normality before applying parametric tests.
   • Sensitive to sample size; small samples may not provide reliable results.

10. Kolmogorov-Smirnov test

    • Compares the empirical distribution function of a sample with a reference probability distribution (e.g., normal distribution).
    • Can be used for one-sample or two-sample tests to assess the goodness of fit.
    • Non-parametric and does not assume a specific distribution for the data.