Hypothesis Testing

Hypothesis testing is a fundamental statistical method used in data science to make inferences about a population based on sample data. It helps in determining whether an observed effect is statistically significant or if it occurred by random chance.

Key Concepts

Null and Alternative Hypotheses

• Null Hypothesis (H₀): A statement that assumes no effect or no difference exists. It represents the default assumption.
• Alternative Hypothesis (H₁ or Ha): A statement that contradicts the null hypothesis, suggesting an effect or a difference exists.

Significance Level (α)

• The probability threshold for rejecting the null hypothesis, commonly set at 0.05 (5%).

P-Value

• A measure of the probability that the observed data would occur if the null hypothesis were true. A small p-value (≤ α) suggests strong evidence against H₀, leading to its rejection.

Test Statistics

• A numerical value calculated from sample data used to determine whether to reject H₀.
• Common test statistics, calculated from hypothesis tests, include:
  • Z-score: Used when population variance is known.
  • T-score: Used when population variance is unknown and the sample size is small.
  • Chi-square statistic: Used for categorical data.
  • F-statistic: Used in variance analysis (ANOVA).

Type I and Type II Errors

• Type I Error (False Positive): Rejecting H₀ when it is actually true.
• Type II Error (False Negative): Failing to reject H₀ when it is actually false.

Steps in Hypothesis Testing

1. State the Hypotheses: The first step is to define the null hypothesis (H₀) and the alternative hypothesis (H₁) clearly.
2. Choose the Significance Level (α): Determine the threshold probability (commonly 0.05) for rejecting the null hypothesis.
3. Select the Appropriate Test: Choose the statistical test that best suits the data and research question, such as a T-test, Chi-square test, or ANOVA.
4. Compute the Test Statistic and P-value: Perform the statistical test to calculate the test statistic and corresponding p-value.
5. Compare P-value with α: If the p-value is less than or equal to α, reject H₀, indicating significant evidence for H₁. Otherwise, fail to reject H₀, suggesting insufficient evidence against it.
6. Draw a Conclusion: Based on the results, interpret whether there is enough evidence to support the alternative hypothesis.

Common Hypothesis Tests

1. Z-Test: Used when the sample size is large (n > 30) and population variance is known.
2. T-Test: Used when the sample size is small (n ≤ 30) and population variance is unknown. Common variants of T-test include:
   • One-sample T-test: Compares sample mean to a known population mean.
   • Two-sample T-test: Compares means of two independent groups.
   • Paired T-test: Compares means of two related groups.
3. Chi-Square Test: Used for categorical data to test the independence or goodness of fit.
4. ANOVA (Analysis of Variance): Typically used when comparing three or more group means.
5. Mann-Whitney U Test: A non-parametric alternative to the T-test for comparing two independent groups.