Tests of Significance

Meaning

• Tests of Significance are statistical tools used to decide whether the observed differences, associations, or relationships in a sample are real (significant) or due to chance (sampling error).
• They help in hypothesis testing by evaluating the probability that the null hypothesis (H₀) is true.

Key Concepts

• Null Hypothesis (H₀): Assumes no difference/association/effect. Example: There is no difference in adoption level between trained and untrained farmers.
• Alternative Hypothesis (H₁): Assumes a real difference/association/effect exists. Example: There is a difference in adoption level between trained and untrained farmers.
• Level of Significance (α): Probability of rejecting H₀ when it is actually true (Type I error). Common values: 0.05 (5%) or 0.01 (1%).
• p-value: The observed probability that test results occurred due to chance. If p ≤ α → reject H₀ (result is significant).
• Types of Errors:
  • Type I Error (α): Rejecting H₀ when it is true.
  • Type II Error (β): Accepting H₀ when it is false.

Types of Tests of Significance

a) Parametric Tests

(Used when data is quantitative, continuous, normally distributed)

• Z-test; For large samples (n > 30). Used to test difference between means or proportions.
• t-test (Student’s t-test)
  • For small samples (n < 30).
  • Types:
    • One-sample t-test → test if sample mean differs from population mean.
    • Independent t-test → compare means of two independent groups.
    • Paired t-test → compare means of same group before & after treatment.
• F-test (ANOVA); Tests difference among three or more means. Example: Comparing mean adoption levels in 3 villages.
• Pearson’s Correlation (r) and Regression; Tests significance of linear relationships.

b) Non-Parametric Tests

(Used when data is ordinal/nominal, or assumptions of parametric tests are not met)

• Chi-Square Test (χ²): Tests association between categorical variables. Example: Association between education and adoption.
• Mann-Whitney U Test: Non-parametric equivalent of independent t-test.
• Wilcoxon Signed-Rank Test: Equivalent of paired t-test.
• Kruskal-Wallis Test: Equivalent of one-way ANOVA.
• Friedman Test: Equivalent of repeated measures ANOVA.

Steps in Test of Significance

• State the null (H₀) and alternative hypothesis (H₁).
• Select the appropriate test (t, z, F, χ², etc.).
• Choose the level of significance (α = 0.05 or 0.01).
• Compute the test statistic using formula.
• Find the critical value from statistical tables.
• Compare test statistic with critical value.
  • If test statistic > critical value → Reject H₀.
  • If test statistic < critical value → Fail to reject H₀.
• Draw conclusion.

Applications in Social Research

• Comparing knowledge gain before and after training (Paired t-test).
• Testing effectiveness of extension teaching methods (ANOVA).
• Checking association between education and adoption (Chi-square).
• Comparing adoption levels of different groups (Mann-Whitney / Kruskal-Wallis).

Key Exam Points

• Tests of significance = hypothesis testing tools.
• Parametric tests → Z, t, F, correlation, regression.
• Non-parametric tests → Chi-square, Mann-Whitney, Wilcoxon, Kruskal-Wallis, Friedman.
• Level of significance: 0.05 (5%) or 0.01 (1%) commonly used.
• Developed by R.A. Fisher, W.S. Gosset (Student’s t-test), etc.