18.1 Hypothesis testing fundamentals

Hypothesis testing is a crucial tool in statistics, allowing us to make informed decisions based on data. It involves comparing a null hypothesis against an alternative hypothesis to determine if observed differences are statistically significant.

The process includes setting up hypotheses, choosing a significance level, collecting data, and calculating test statistics. By following these steps, we can draw meaningful conclusions about populations from sample data, guiding decision-making in various fields.

Hypothesis Testing Fundamentals

Null vs alternative hypotheses

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• Null hypothesis ($H_0$) represents the default or status quo claim assumes no significant difference or effect usually includes an equality (=, ≤, or ≥)
• Alternative hypothesis ($H_a$ or $H_1$) represents the claim that contradicts the null hypothesis suggests a significant difference or effect usually includes an inequality (≠, >, or <)
• Examples:
  • Testing a new medication's effectiveness compared to a standard treatment
    • $H_0$: The new medication is no more effective than the standard treatment
    • $H_a$: The new medication is more effective than the standard treatment
  • Investigating the impact of a new teaching method on student performance
    • $H_0$: The new teaching method does not improve student performance
    • $H_a$: The new teaching method improves student performance

Purpose of hypothesis testing

• Make data-driven decisions about population parameters based on sample statistics determine if observed differences are statistically significant or due to chance
• Hypothesis testing allows researchers to test claims or theories about a population by analyzing sample data provides a structured approach to making inferences and drawing conclusions
• Examples:
  • Determining if a new product feature increases customer satisfaction
  • Investigating if a certain factor (age, gender) influences consumer behavior

Critical region and significance level

• Significance level ($\alpha$) is the probability of rejecting the null hypothesis when it is true (Type I error) commonly used values are 0.01, 0.05, or 0.10
• Critical region is the range of values for the test statistic that leads to rejecting the null hypothesis determined by the significance level and the type of test (one-tailed or two-tailed)
  • One-tailed tests: Upper-tailed test has critical region in the right tail of the distribution Lower-tailed test has critical region in the left tail of the distribution
  • Two-tailed test: Critical region is divided equally between the left and right tails of the distribution
• The choice of significance level depends on the consequences of a Type I error (rejecting a true null hypothesis) a smaller $\alpha$ reduces the chances of a Type I error but increases the chances of a Type II error (failing to reject a false null hypothesis)

Steps in hypothesis testing

1. State the null and alternative hypotheses: Clearly define $H_0$ and $H_a$ based on the problem statement
2. Choose a significance level ($\alpha$): Select an appropriate value based on the consequences of a Type I error
3. Collect sample data: Gather relevant data through experiments, surveys, or observations
4. Calculate the test statistic: Use the appropriate formula based on the type of test and data (z-test, t-test, chi-square test)
5. Determine the critical value or p-value:
   • Find the critical value using the significance level and the appropriate distribution
   • Calculate the p-value using the test statistic and the appropriate distribution
6. Compare the test statistic to the critical value or p-value:
   • If using the critical value approach, reject $H_0$ if the test statistic falls in the critical region
   • If using the p-value approach, reject $H_0$ if the p-value is less than the significance level
7. Make a decision and interpret the results: State whether to reject or fail to reject the null hypothesis interpret the results in the context of the original problem