Module 22 Wrap-Up: Hypothesis Test for Population Mean

Overview

Focus on hypothesis testing for a population mean.
Involves four key steps:
1. Determine the hypotheses.
2. Collect the data.
3. Assess the evidence.
4. Draw a conclusion.

Null Hypothesis (H₀): Claims the mean (μ) equals a specific value (μ₀).
Alternative Hypothesis (H₁): Competing claim that μ is less than, greater than, or not equal to μ₀.
- Testing μ < μ₀ or μ > μ₀ are one-sided tests.
- Testing μ ≠ μ₀ is a two-sided test.
For matched pairs designs, focus on the difference in two measurements and μ is the mean of these differences.

Hypothesis tests rely on probability; random selection or assignment is essential.
Check if the t-model fits:
- Variable must be normally distributed or sample size > 30.
- If not verifiable and sample size < 30, use the t-model if the sample isn’t strongly skewed and lacks outliers.

If using t-model, determine the test statistic:
- Formula: [ t = \frac{\bar{x} - μ}{s / \sqrt{n}} ]
Use the test statistic and alternative hypothesis to find the P-value:
- P-value: Probability of finding a sample mean as extreme as the observed, assuming H₀ is true.
- If H₁ is >, P-value is area right of the test statistic.
- If H₁ is <, P-value is area left of the test statistic.
- If H₁ is ≠, P-value is double the tail area beyond the test statistic.

Compare P-value to significance level (α):
- If P ≤ α: Reject H₀, conclude significant evidence for H₁.
- If P > α: Fail to reject H₀, conclude insufficient evidence for H₁.
Conclusion should relate back to the research question and include the P-value.

P-value: Probability related to the sample data assuming null hypothesis is true.
Errors:
- Type I Error: Rejecting a true null hypothesis.
- Type II Error: Failing to reject a false null hypothesis.
To avoid Type I errors in critical cases, use a stricter significance level (e.g., α = 0.01).
"Garbage in, garbage out": Poor data collection leads to meaningless test results.