📊 Sessions 9.1.5, 9.1.6 - Hypothesis Tests & Confidence Intervals

📌 Sessions 9.1.5, 9.1.6 - Hypothesis Tests & Confidence Intervals

Hypothesis testing and confidence intervals are closely related. A 100(1 – α)% confidence interval for μ will lead to rejection of H₀: μ = μ₀ at significance level α if μ₀ is not inside the interval.

📎 Example: For a medicine test, suppose we test H₀: μ = 50 using x̄ = 51.3, σ = 2.5, and n = 16 at α = 0.05. The 95% CI is:

51.3 ± 1.96 × (2.5 / √16) → [50.075, 52.525]

Since μ₀ = 50 is not in this interval, we reject H₀. Thus, the CI and hypothesis test lead to the same decision.

Statistical significance: P-value is small → reject H₀.
Practical significance: The effect size (e.g., μ = 50.5 vs. 50) may be too small to matter in practice.

Conclusion: Always use both the P-value and confidence interval to interpret results and assess real-world importance.

📊 Practical vs. Statistical Significance

This example demonstrates how increasing sample size can lead to rejecting the null hypothesis (H₀), even when the practical difference is negligible.

🧪 Scenario

We are testing whether the mean burning rate of a propellant differs from 50 cm/sec.

📋 Results Table

Sample Size (n)	Z-score	P-value	Reject H₀?	Practical Difference
10	0.63	0.528	No	No
25	1.00	0.317	No	No
100	2.00	0.046	Yes	Still minimal
1000	6.32	2.5×10⁻¹⁰	Yes	Still minimal

🔢 7-Step Hypothesis Testing Framework

Parameter of interest: Identify from the problem context.
Null hypothesis H₀: State the claim to test.
Alternative hypothesis H₁: Define the opposing claim (directional or non-directional).
Test statistic: Choose based on data and assumptions (e.g., Z, t).
Reject H₀ if...: Define critical region or α-level threshold.
Computations: Plug in sample data to compute the test statistic.
Draw conclusions: Reject or fail to reject H₀ and interpret in context.

This structured approach ensures careful, logical decision-making, especially when first learning about hypothesis testing.