9.1 Null and Alternative Hypotheses

Hypothesis testing is a powerful tool for making decisions based on data. It involves setting up null and alternative hypotheses, then using statistical methods to determine which is more likely to be true.

The process includes formulating hypotheses, interpreting symbols, and making decisions based on test statistics or p-values. Understanding this process is crucial for drawing meaningful conclusions from data in various fields.

Hypothesis Testing

Formulation of statistical hypotheses

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• Null hypothesis ($H_0$) asserts no significant difference or effect exists
  • Represents the default or status quo position (no change from current understanding)
  • Often includes an equality symbol (=, ≤, or ≥) (population mean $\mu = 100$)
• Alternative hypothesis ($H_a$ or $H_1$) asserts a significant difference or effect exists
  • Represents the research question or claim being tested (new drug is effective)
  • Often includes an inequality symbol (≠, >, or <) (population proportion $p > 0.5$)
• Examples:
  • Testing a new teaching method: $H_0$: No improvement in test scores, $H_a$: Improvement in test scores
  • Comparing product defect rates: $H_0$: Defect rates are equal ($p_1 = p_2$), $H_a$: Defect rates differ ($p_1 \neq p_2$)

Interpretation of hypothesis symbols

• $H_0$ denotes the null hypothesis
• $H_a$ or $H_1$ denotes the alternative hypothesis
• Greek letters represent population parameters:
  • $\mu$ represents the population mean (average value)
  • $p$ represents the population proportion (fraction or percentage)
  • $\sigma$ represents the population standard deviation (measure of variability)
• Subscripts differentiate between specific populations or groups ($\mu_1$ vs. $\mu_2$)
• Equality symbols in $H_0$ suggest no difference or effect (= for exactly equal, ≤ or ≥ for at most or at least)
• Inequality symbols in $H_a$ suggest a difference or effect (≠ for not equal, > or < for greater than or less than)

Decision-making in hypothesis testing

1. Gather sample data relevant to the hypotheses being tested
2. Compute the appropriate test statistic based on the data and hypothesis (z-score for normal data, t-score for small samples, chi-square for categorical data)
3. Determine the critical value(s) using the chosen significance level ($\alpha$, often 0.05)
   • If the test statistic is in the critical region (more extreme than the critical value), reject $H_0$
   • If the test statistic is outside the critical region (less extreme than the critical value), fail to reject $H_0$
4. Alternatively, calculate the p-value (probability of observing the sample data or more extreme results, assuming $H_0$ is true)
   • If the p-value is less than $\alpha$, reject $H_0$ (statistically significant result)
   • If the p-value is greater than or equal to $\alpha$, fail to reject $H_0$ (not statistically significant)
5. Draw conclusions based on the decision regarding $H_0$
   • Rejecting $H_0$ suggests the sample data supports $H_a$ (evidence for a significant difference or effect)
   • Failing to reject $H_0$ suggests insufficient evidence to support $H_a$ (cannot conclude a significant difference or effect exists)

Statistical Inference and Interpretation

• Statistical inference involves drawing conclusions about populations based on sample data
• Hypothesis testing is a key method in statistical inference for making decisions about population parameters
• Statistical significance indicates the likelihood that an observed effect is not due to chance
• Confidence intervals provide a range of plausible values for population parameters, complementing hypothesis tests
• Effect size measures the magnitude of the observed difference or relationship, providing context for statistical significance