Concept of Hypothesis Testing
Hypothesis testing is a statistical method used to make decisions or inferences about a population based on sample data. It helps in determining whether a statement or assumption about a parameter (e.g., mean, proportion) is valid.
Key Terminology in Hypothesis Testing
- Null Hypothesis (H₀): A statement of no effect or no difference.
- Alternative Hypothesis (H₁): A statement indicating a significant effect or difference.
- Type I Error (α): Rejecting a true null hypothesis (false positive).
- Type II Error (β): Failing to reject a false null hypothesis (false negative).
- P-value: The probability of obtaining the observed results under the assumption that H₀ is true.
Level of Significance
The level of significance (α) is the probability threshold below which the null hypothesis is rejected. Common significance levels are:
- 0.05 (5%) – Most commonly used.
- 0.01 (1%) – Used for more stringent testing.
- 0.10 (10%) – Used in exploratory research.
A smaller α means stronger evidence is needed to reject H₀.
Process of Hypothesis Testing
- State the Hypotheses: Define H₀ and H₁.
- Select the Significance Level (α): Choose an appropriate threshold.
- Choose the Appropriate Test: Decide between parametric and non-parametric tests.
- Calculate the Test Statistic: Compute the relevant statistic based on sample data.
- Determine the Critical Value or P-value: Compare the test statistic to the critical value or compute the p-value.
- Make a Decision:
- If p-value < α, reject H₀ (significant result).
- If p-value ≥ α, fail to reject H₀ (not significant).
- Interpret the Results: Draw conclusions and provide insights based on findings.
Test of Hypothesis Concerning Mean
Testing hypotheses about a population mean involves comparing a sample mean with a known or hypothesized population mean.
1. Normal Z-Test for Single Mean
Used when:
- Sample size is large (n ≥ 30).
- Population standard deviation (σ) is known.
Formula for Z-test:
Z = (X̄ – μ) / (σ / √n)
where:
- X̄ = Sample mean
- μ = Population mean
- σ = Population standard deviation
- n = Sample size
2. Student’s t-Test for Single Mean
Used when:
- Sample size is small (n < 30).
- Population standard deviation is unknown.
Formula for t-test:
t = (X̄ – μ) / (s / √n)
where:
- s = Sample standard deviation
The t-test follows a t-distribution with (n-1) degrees of freedom.
Using Non-Parametric Statistics for Hypothesis Testing
Non-parametric tests are used when data do not meet the assumptions of normality and homogeneity of variance.
Common Non-Parametric Tests:
- Mann-Whitney U Test: Used to compare two independent samples when normality is not assumed.
- Wilcoxon Signed-Rank Test: Used for comparing paired data when the normality assumption is violated.
- Kruskal-Wallis Test: Non-parametric equivalent of ANOVA, used for comparing more than two groups.
- Chi-Square Test: Used for categorical data to test associations between variables.
Conclusion
Hypothesis testing is a fundamental statistical tool for decision-making. By selecting the appropriate test—whether parametric (Z-test, t-test) or non-parametric (Mann-Whitney, Wilcoxon, Kruskal-Wallis)—researchers can validate hypotheses and make data-driven conclusions.