9.5 Additional Information and Full Hypothesis Test Examples

Hypothesis testing is a crucial statistical method for making decisions about populations based on sample data. It involves setting up hypotheses, calculating test statistics, and interpreting p-values to draw conclusions about the validity of claims.

The process includes formulating null and alternative hypotheses, determining significance levels, and calculating test statistics. By comparing p-values to predetermined thresholds, researchers can make informed decisions about rejecting or failing to reject null hypotheses, guiding further actions or investigations.

Hypothesis Testing Concepts

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Significance and p-value interpretation

• Level of significance ($\alpha$) represents the maximum acceptable risk of rejecting the null hypothesis ($H_0$) when it is actually true (Type I error)
  • Commonly set at 0.01, 0.05, or 0.10 and determined before conducting the test
• p-value quantifies the probability of obtaining a test statistic as extreme as or more extreme than the observed value, assuming $H_0$ is true
  • Calculated from sample data and compared to $\alpha$ to make a decision
    • Reject $H_0$ if p-value ≤ $\alpha$ (statistically significant)
    • Fail to reject $H_0$ if p-value > $\alpha$ (not statistically significant)
• Role in decision-making: $\alpha$ sets the threshold for rejecting $H_0$, while p-value provides evidence against $H_0$
  • Smaller p-values indicate stronger evidence against $H_0$ (0.001 vs 0.04)
• Statistical power is the probability of correctly rejecting $H_0$ when it is false, which increases with larger effect sizes

Types of hypothesis tests

• Alternative hypothesis ($H_a$ or $H_1$) determines the test type based on the parameter and value
  • Left-tailed: $H_a: \text{parameter} < \text{value}$ with critical region in left tail ($H_a: \mu < 100$)
  • Right-tailed: $H_a: \text{parameter} > \text{value}$ with critical region in right tail ($H_a: p > 0.5$)
  • Two-tailed: $H_a: \text{parameter} \neq \text{value}$ with critical region split between left and right tails ($H_a: \mu \neq 50$)
• Critical value(s) depend on test type and $\alpha$
  • Left-tailed: Lower critical value
  • Right-tailed: Upper critical value
  • Two-tailed: Lower and upper critical values ($-z_{\alpha/2}$ and $z_{\alpha/2}$ for z-test)

Test Planning and Interpretation

• Confidence interval provides a range of plausible values for the population parameter, complementing hypothesis test results
• Effect size quantifies the magnitude of the difference between groups or the strength of a relationship
• Sample size determination is crucial for achieving desired statistical power and precision in hypothesis testing

Full Hypothesis Test Example

Single population proportion test

1. Formulate null and alternative hypotheses

   • $H_0: p = p_0$ ($p_0$ is claimed proportion)
   • $H_a: p < p_0$, $p > p_0$, or $p \neq p_0$ based on context

2. Determine significance level ($\alpha$)

3. Calculate test statistic ($z$) using sample proportion ($\hat{p}$)

   • $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$ ($n$ is sample size)

4. Find p-value based on test statistic and test type using standard normal distribution

5. Compare p-value to $\alpha$

   • Reject $H_0$ if p-value ≤ $\alpha$
   • Fail to reject $H_0$ if p-value > $\alpha$

6. Interpret results in problem context

   • State if there is sufficient evidence to support $H_a$
   • Discuss practical implications of findings (election outcome, product defect rate)