| Test name | Purpose | Example Study in Social Sciences | Type of Hypothesis Tested | Hypothesis Model | Interpretation Guidelines | Assumptions | When to Use | Formula |
| One Sample t Test | Assessing Mean Differences Between Sample and Population | Analyzing IQ Scores of Students Against Population Mean | One-tailed or Two-tailed | H0:μ=μ0 Ha:μ≠μ0 | Reject H0 if p<α | Data is normally distributed, Population variance is unknown | When comparing a sample mean to a known or hypothesized population mean | t=nsxˉ−μ0 |
| Two Sample t Test | Comparing Means of Two Independent Samples | Evaluating Test Scores of Students Under Different Teaching Methods | Two-tailed | H0:μ1=μ2 Ha:μ≠μ2 | Reject H0 if p<α | Data is normally distributed, Homogeneity of variances | When comparing means of two independent samples | t=n1s12+n2s22xˉ1−xˉ2 |
| Paired Sample t Test | Determining Mean Differences Within Paired Data | Examining Pre-test and Post-test Scores of Same Group of Students | Two-tailed | H0:μd=0 Ha:μd≠0 | Reject H0 if p<α | Data is normally distributed, Paired observations | When comparing means of paired samples | t=nsddˉ |
| Single Sample Proportion | Testing Population Proportion Against Hypothesized Proportion | Investigating Voter Support Proportion for a Political Candidate | One-tailed or Two-tailed | H0:p=p0 Ha:p≠p0 | Reject H0 if p<α | Large sample size | When comparing a sample proportion to a hypothesized proportion | z=np0(1−p0)p^−p0 |
| Two Sample Proportion | Comparing Proportions Between Two Independent Samples | Analyzing Product Preference Proportions Among Genders | Two-tailed | H0:p1=p2 Ha:p1≠p2 | Reject H0 if p<α | Large sample size | When comparing proportions of two independent samples | z=p(1−p)(n11+n21)(p^1−p^2) |
| Proportions in Several Categories | Comparing Proportions Across Multiple Categories | Investigating Political Party Preferences Among Different Age Groups | Two-tailed | H0:p1=p2=…=pk Ha:At least one pi=pj | Reject H0 if p<α | Large sample size, Independent observations | When comparing proportions across multiple categories | X2=∑Ei(Oi−Ei)2 |
| Chi Square Goodness of Fit | Testing Whether Observed Frequencies Match Expected Frequencies | Assessing Whether Observed Distribution of Political Affiliation Matches Expected Distribution | Two-tailed | H0:Observed frequencies match expected frequencies Ha:Observed frequencies do not match expected frequencies | Reject H0 if p<α | Random sampling | When comparing observed frequencies to expected frequencies | X2=∑Ei(Oi−Ei)2 |
| Chi Square Test for Association | Assessing Whether Two Categorical Variables Are Independent | Examining the Relationship Between Gender and Voting Preference | Two-tailed | H0:No association between variables Ha:Association between variables | Reject H0 if p<α | Random sampling | When assessing independence between two categorical variables | X2=∑Ei(Oi−Ei)2 |
| Wilcoxon Signed-Rank Test | Comparing Distributions of Two Paired Samples | Investigating Changes in Pain Perception Before and After Treatment | Two-tailed | H0:No difference in medians Ha:At least one median differs | Reject H0 if p<α | Random sampling | When comparing two related samples | W=sum of ranks of positive differences |
| Mann-Whitney U Test | Comparing Distributions of Two Independent Samples | Analyzing Test Scores of Students Between Two Different Schools | Two-tailed | H0:No difference in medians Ha:At least one median differs | Reject H0 if p<α | Independent observations, Continuous or ordinal data | When comparing two independent samples | |
| McNemar’s Test | Assessing Changes in Binary Outcomes Over Time | Examining Changes in Smoking Habits Before and After Intervention | Two-tailed | H0:No difference in proportions Ha:At least one proportion differs | Reject H0 if p<α | Dependent observations, Binary data | When comparing binary outcomes from paired observations | X2=b+c(b−c)2 |
| Kruskal-Wallis Test | Comparing Distributions of Three or More Independent Samples | Analyzing Test Scores of Students Across Multiple Schools | Two-tailed | H0:No difference in medians Ha:At least one median differs | Reject H0 if p<α | Independent observations, Continuous or ordinal data | When comparing three or more independent samples | Rank sums of groups are compared |
| Friedman Test | Assessing Differences in Related Samples Across Multiple Groups | Evaluating Student Performance Across Different Exam Formats | Two-tailed | H0:No difference in medians Ha:At least one median differs | Reject H0 if p<α | Dependent observations, Continuous or ordinal data | When comparing related samples across multiple groups | Rank sums of groups are compared |
| Scheffé Test | Comparing Specific Mean Differences Between Multiple Groups | Analyzing Test Scores of Students Across Various Teaching Methods | Two-tailed | H0:No difference in means Ha:At least one mean differs | Reject H0 if p<α | Homogeneity of variances, Large sample size | When comparing specific mean differences between multiple groups | t=MSE×(n11+n21)xˉ1−xˉ2 |
| Permutation Test | Nonparametric Test for Comparing Distributions | Assessing Difference in Height Between Two Groups of Trees | Two-tailed | H0:No difference in distributions Ha:Difference in distributions | Reject H0 if p<α | Random sampling, Independent observations | When comparing distributions without assuming underlying distribution | Calculate the observed test statistic and simulate many random permutations of the data to create a null distribution |
| Kruskal-Wallis Test | Comparing Distributions of Three or More Independent Samples | Analyzing Test Scores of Students Across Multiple Schools | Two-tailed | H0:No difference in medians Ha:At least one median differs | Reject H0 if p<α | Independent observations, Continuous or ordinal data | When comparing three or more independent samples | Rank sums of groups are compared |
| One-way ANOVA | Comparing Means of Three or More Independent Groups | Examining the Effect of Different Teaching Methods on Student Performance | Two-tailed | H0:No difference in means Ha:At least one mean differs | Reject H0 if p<α | Homogeneity of variances, Normally distributed data | When comparing means of three or more independent groups | WithinF=MSWithinMSBetween |
| Two-way ANOVA | Comparing Means Across Two Independent Variables | Analyzing the Effect of Gender and Teaching Method on Student Performance | Two-tailed | H0:No interaction effect Ha:Interaction effect | Reject H0 if p<α | Homogeneity of variances, Normally distributed data | When examining the combined effects of two independent variables | ErrorF=MSErrorMSFactor A |
| Multifactor ANOVA | Assessing the Effects of Multiple Independent Variables | Investigating the Impact of Socioeconomic Status, Ethnicity, and Education Level on Academic Achievement | Two-tailed | H0:No interaction effet Ha:Interaction effect | Reject H0 if p<α | Homogeneity of variances, Normally distributed data | When studying the combined effects of multiple independent variables | ErrorF=MSErrorMSFactor A |
| Factorial ANOVA | Analyzing the Interaction Between Factors and Their Main Effects | Exploring the Influence of Temperature and Humidity on Plant Growth | Two-tailed | H0:No interaction effet Ha:Interaction effect | Reject H0 if p<α | Homogeneity of variances, Normally distributed data | When studying the interaction between factors and their main effects | ErrorF=MSErrorMSFactor A |
| Pearson Correlation | Examining the Strength and Direction of a Linear Relationship | Investigating the Relationship Between Study Hours and Exam Scores | Two-tailed | H0:No correlation Ha:Correlation | Reject H0 if p<α | Linear relationship between variables | When assessing the linear association between two continuous variables | r=[n(∑x2)−(∑x)2][n(∑y2)−(∑y)2]n(∑xy)−(∑x)(∑y) |
| Bland-Altman Correlation | Assessing Agreement Between Two Measurement Techniques | Comparing Blood Pressure Measurements Between Two Devices | Two-tailed | H0:No difference in measurements Ha:Difference in measurements | Reject H0 if p<α | Linear relationship between variables | When evaluating agreement between two measurement techniques | Plot the difference between measurements against their mean to assess agreement |
| Spearman Rank Correlation | Assessing the Strength and Direction of a Monotonic Relationship | Investigating the Relationship Between Rank-Ordered Variables | Two-tailed | H0:No correlation Ha:Correlation | Reject H0 if p<α | Monotonic relationship between variables | When assessing the relationship between two ordinal or ranked variables | rs=1−n(n2−1)6∑d2 |
| Simple Linear Regression | Predicting the Value of a Dependent Variable Based on One Predictor | Predicting Exam Scores Based on Study Hours | Two-tailed | H0:No relationship Ha:Relationship | Reject H0 if p<α | Linear relationship between variables | When predicting the value of a dependent variable based on one predictor | Y=β0+β1X+ϵ |
| Logistic Regression | Predicting the Probability of a Binary Outcome | Predicting the Likelihood of Students Passing an Exam Based on Study Hours | Two-tailed | H0:No relationship Ha:Relationship | Reject H0 if p<α | Linearity, Independence, No multicollinearity | When predicting the probability of a binary outcome | P(Y=1)=1+e−(β0+β1X)1 |
| Loglinear Models | Analyzing the Relationship Between Categorical Variables | Examining the Association Between Socioeconomic Status and Education Level | Two-tailed | H0:No relationship Ha:Relationship | Reject H0 if p<α | No multicollinearity | When analyzing the relationship between multiple categorical variables | Loglinear equations are fitted to the data and likelihood ratio tests are used to compare models |
| Factor Analysis | Identifying Patterns in Data and Reducing Dimensionality | Investigating the Structure of Questionnaire Items on Student Satisfaction | Two-tailed | H0:No relationship Ha:Relationship | Reject H0 if p<α | Factorability, No multicollinearity | When exploring patterns in data and reducing dimensionality | Factors are extracted from the correlation matrix and rotated to simplify interpretation |
| Cluster Analysis | Identifying Groups or Clusters Within a Dataset | Segmenting Customers Based on Buying Behavior | Two-tailed | H0:No relationship Ha:Relationship | Reject H0 if p<α | Homogeneity within clusters, Heterogeneity between clusters | When identifying natural groupings or clusters within a dataset | Distance measures are used to calculate similarities between cases, which are then grouped into clusters |
| Principal Component | Reducing the Dimensionality of Data and Identifying Patterns | Analyzing Variability in Student Performance Based on Multiple Test Scores | Two-tailed | H0:No relationship Ha:Relationship | Reject H0 if p<α | Linear relationship between variables | When reducing the dimensionality of data and identifying underlying patterns | Principal components are extracted from the correlation or covariance matrix and rotated for interpretation |
| Discriminant Analysis | Predicting Group Membership Based on Predictor Variables | Predicting Student Enrollment in Advanced Placement Courses | Two-tailed | H0:No difference in group means Ha:Difference in group means | Reject H0 if p<α | Normality, Homogeneity of Covariance Matrices | When predicting group membership based on predictor variables | Discriminant functions are computed to classify cases into groups based on predictor variables |
