Hypothesis Tests for the Mean (μ)

Key Concepts

• Parameter: Represents an unknown population value.
• Statistic: Calculated from a sample to estimate the parameter.
• Confidence Intervals: Used previously to estimate population means.
  • Example: 95% confidence interval should contain the population mean 95% of the time.
• Hypothesis Testing: Evaluates a claim about a population parameter using sample data.

Steps in Hypothesis Testing

Requirements (General Assumptions)

1. Simple Random Sample: The sample must be randomly chosen from the population.
2. Normal Distribution: Population parameter follows a normal distribution (μ, σ).
3. Known Standard Deviation (σ): Population standard deviation is known, though this assumption will be relaxed in future chapters.

Hypothesis Testing Process

1. Formulating the Hypotheses

   • Null Hypothesis (H₀): Assumes the claim (e.g., μ = 50) is true.
   • Alternate Hypothesis (H₁): Represents the opposite of the null (can be two-sided or one-sided).
     • Two-sided: μ ≠ μ₀
     • One-sided: μ > μ₀ or μ < μ₀

2. Setting the Significance Level (α)

   • Common choice: α = 0.05, meaning a 5% chance of rejecting H₀ when it is true.
   • Corresponds to z-value: ±1.96 for a 95% confidence interval.

3. Collecting Data and Performing Calculations

   • Z-test Statistic Calculation
     • Formula: (Observed - Expected) / Spread
     • Example: Using sample mean (x̅), population mean (μ), and standard deviation (σ).

4. Making a Conclusion

   • Compare z-test statistic to z-value threshold.
     • |z-test| > z-star: Reject H₀ (the result is statistically significant and unlikely under H₀).
     • |z-test| ≤ z-star: Do not reject H₀ (the result is not statistically significant).
   • Analogy: Legal system presumption of innocence until proven guilty.

Example Scenario

• Claim: Average height (μ) of Point Loma Nazarene University (PLNU) students is 69 inches.
• Sample Data: 25 students, sample mean (x̅) = 67.3 inches, σ = 2.65.
• Steps:
  1. Null Hypothesis (H₀): μ = 69 inches.
  2. Alternate Hypothesis (H₁): μ ≠ 69 inches.
  3. Significance Level (α): 0.05 (z-value = ±1.96).
  4. Calculate z-test Statistic: z = (67.3 - 69) / (2.65 / √25) = -3.208.
  5. Conclusion: |z| > 1.96, thus reject H₀ (unlikely that μ = 69 inches).

Key Takeaways

• Hypothesis testing is a step-by-step process to evaluate population claims using sample data.
• The null hypothesis is assumed true until sample evidence suggests otherwise.
• Ethical statistical practice involves setting hypotheses and significance levels before data collection.
• Most real-world examples, like product claims and medical dosages, hinge on this statistical method.

Further Discussion

• Ethical considerations and proper sequence in hypothesis testing.
• Real-life examples where hypothesis testing is applicable.
• In-class exercises to practice calculations and interpretations of z-test statistics.

Additional Questions

• How do variations in sample size affect the z-test statistic?
• What happens if σ is unknown?
• How do we interpret results in practical scenarios?