Hypothesis Testing in Statistics

What Is Hypothesis Testing?

• A structured method to determine if study findings support a specific theory relevant to a larger population.
• Used to test assumptions about population parameters and estimate relationships between statistical variables.
• Requires understanding different data types like nominal data.

Examples of Hypothesis Testing

• A teacher assumes 60% of students are from lower-middle-class families.
• A doctor believes a regimen is 90% effective for diabetic patients.

Importance of Hypothesis Testing

• Avoiding Misleading Conclusions: Helps avoid Type I and Type II errors by setting significance levels and calculating the p-value.
• Making Smarter Choices: Allows for evidence-based decisions, e.g., testing if a new park increases community engagement.
• Optimizing Business Tactics: Tests new business strategies before full implementation, e.g., free shipping offer's impact on sales.

Hypothesis Testing Formula

• Z = (x - μ) / (σ/√n)
  • x: sample mean
  • μ: population mean
  • σ: standard deviation
  • n: sample size

How Hypothesis Testing Works

• Tests two hypotheses: null (H0) and alternative (H1).
• Null hypothesis assumes no effect; alternative is the opposite.
• Only one of the hypotheses can be correct.

Null and Alternative Hypothesis

• H0 (Null Hypothesis): Event will not happen, e.g., average return equals zero.
• H1 (Alternative Hypothesis): Opposite of H0, e.g., average return is not zero.

Example

• A sanitizer claims to kill 95% of germs.
  • H0: Average = 95%
  • H1: Average < 95%

Calculation Example

• Test average height of women in the US: Null hypothesis is average height = 5'4".
• Sample of 100 women shows an average height of 5'5".
• Calculate z-score: 11.11, reject H0.

Steps in Hypothesis Testing

1. Formulate Hypotheses: Define H0 and H1.
2. Choose Significance Level (α): Common values are 0.05, 0.01, 0.10.
3. Select the Test: Based on data type, sample size, etc.
4. Collect Data: Gather representative data.
5. Calculate Test Statistic: Reflects deviation from H0.
6. Determine p-value: Probability of observing results if H0 is true.
7. Make a Decision: Compare p-value to α.
8. Report Results: Present findings and conclusion.
9. Post-hoc Analysis: Further analysis if needed.

Types of Hypothesis Testing

• Z Test: Tests if two means are the same (population standard deviation known, n ≥ 30).
• T Test: Compares means of two groups.
• Chi-Square: Tests if observed data fits expected data.
• ANOVA: Compares means of three or more groups.

Hypothesis Testing and Confidence Intervals

• Confidence intervals estimate population parameters.
• Hypothesis tests examine given hypotheses.
• Results from both are typically consistent at given significance levels.

Simple and Composite Hypothesis

• Simple Hypothesis: Specifies an exact parameter value.
• Composite Hypothesis: Specifies a range of values.

Example

• Simple: Average sales = 1000 units.
• Composite: Sales range from 900 to 1000 units.

One-Tailed and Two-Tailed Testing

• One-Tailed: Tests for greater or lesser than a value.
• Two-Tailed: Tests for differences in both directions.

Type 1 and Type 2 Errors

• Type 1 Error: Rejecting a true null hypothesis.
• Type 2 Error: Failing to reject a false null hypothesis.

Applications of Hypothesis Testing

• Medicine and Healthcare: Testing new treatments.
• Quality Control: Maintaining product standards.
• Education: Evaluating teaching methods.
• Environmental Science: Assessing conservation measures.
• Marketing: Refining strategies.

Limitations of Hypothesis Testing

• Cannot prove truth; results are sample-specific.
• Prone to errors and requires assumptions.

Conclusion

• Hypothesis testing is essential in data science and helps make informed decisions based on statistical evidence.