8.1 Fundamentals of Hypothesis Testing

Hypothesis testing is a crucial tool in statistics, allowing us to make decisions about populations based on sample data. It involves formulating null and alternative hypotheses, then using statistical methods to determine if there's enough evidence to reject the null hypothesis.

The process includes setting up hypotheses, choosing a significance level, calculating test statistics, and interpreting results. Understanding these steps is key to making informed decisions in various fields, from medical research to business analytics.

Hypothesis Testing Fundamentals

Purpose of hypothesis tests

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• Assess claims or hypotheses about population parameters using sample evidence
• Determine if observed differences or effects are statistically significant or due to chance
• Examples:
  • Testing if a new drug is more effective than a placebo
  • Examining if there is a significant difference in mean scores between two groups

Null vs alternative hypotheses

• Null hypothesis ($H_0$)
  • Assumes no effect, difference, or relationship exists in the population
  • Includes equality (=, ≤, or ≥) and represents the status quo
  • Rejected only when strong evidence against it is present
  • Example: $H_0$: The mean weight loss of the new diet is equal to or less than the current diet
• Alternative hypothesis ($H_a$ or $H_1$)
  • Contradicts the null hypothesis and represents the research claim or expected difference
  • Includes inequality (<, >, or ≠)
  • Accepted when the null hypothesis is rejected
  • Example: $H_a$: The mean weight loss of the new diet is greater than the current diet

Steps in hypothesis testing

1. Formulate the null and alternative hypotheses based on the research question
2. Select the appropriate test statistic and its distribution under the null hypothesis
3. Specify the significance level ($\alpha$) as the probability threshold for rejecting $H_0$ when true
   • Common choices: 0.05 or 0.01
4. Compute the test statistic value using the sample data
5. Find the p-value or critical value(s) associated with the test statistic
   • p-value: Probability of observing a test statistic as extreme as or more extreme than the calculated one, assuming $H_0$ is true
   • Critical value(s): Boundary value(s) that separates the rejection and non-rejection regions based on the significance level
6. Decide to reject or fail to reject $H_0$ by comparing the p-value to $\alpha$ or the test statistic to the critical value(s)
7. Interpret the results in the context of the original problem, considering the implications and limitations

Interpretation of test results

• Rejecting the null hypothesis
  • Sufficient evidence supports the alternative hypothesis
  • Findings are statistically significant at the chosen $\alpha$ level
  • Example: Rejecting $H_0$ suggests the new drug is more effective than the placebo
• Failing to reject the null hypothesis
  • Insufficient evidence to support the alternative hypothesis
  • Findings are not statistically significant at the chosen $\alpha$ level
  • Example: Failing to reject $H_0$ indicates no significant difference in mean scores between groups
• Interpreting results in the problem's context
  • Relate findings to the study's objectives and implications
  • Discuss potential sources of error, bias, or limitations affecting the outcome
  • Example: A significant result may suggest implementing the new diet, but long-term effects and adherence should be considered