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

1. State the Hypotheses: Define H₀ and H₁.
2. Select the Significance Level (α): Choose an appropriate threshold.
3. Choose the Appropriate Test: Decide between parametric and non-parametric tests.
4. Calculate the Test Statistic: Compute the relevant statistic based on sample data.
5. Determine the Critical Value or P-value: Compare the test statistic to the critical value or compute the p-value.
6. Make a Decision:
   • If p-value < α, reject H₀ (significant result).
   • If p-value ≥ α, fail to reject H₀ (not significant).
7. 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:

1. Mann-Whitney U Test: Used to compare two independent samples when normality is not assumed.
2. Wilcoxon Signed-Rank Test: Used for comparing paired data when the normality assumption is violated.
3. Kruskal-Wallis Test: Non-parametric equivalent of ANOVA, used for comparing more than two groups.
4. 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.