12.1 Statistical inference and hypothesis testing

Statistical inference and hypothesis testing are crucial tools in experimental design. They allow researchers to draw conclusions about populations based on sample data. These methods help determine if observed effects are statistically significant or simply due to chance.

Hypothesis testing involves formulating null and alternative hypotheses, setting significance levels, and calculating p-values. Understanding Type I and Type II errors, as well as statistical power, is essential for interpreting results accurately and drawing valid conclusions from experiments.

Hypothesis Testing

Formulating Hypotheses

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• Null hypothesis ($H_0$) represents the default position or the assumption that there is no significant effect or difference
• Alternative hypothesis ($H_a$ or $H_1$) represents the claim that contradicts the null hypothesis and suggests a significant effect or difference exists
• Significance level ($\alpha$) is the probability threshold for rejecting the null hypothesis, commonly set at 0.05 (5%) or 0.01 (1%)
• One-tailed test examines the alternative hypothesis in only one direction (greater than or less than the null hypothesis value)
• Two-tailed test considers the alternative hypothesis in both directions (not equal to the null hypothesis value)

Evaluating Hypotheses

• p-value represents the probability of obtaining the observed results or more extreme results, assuming the null hypothesis is true
  • If the p-value is less than the chosen significance level, the null hypothesis is rejected in favor of the alternative hypothesis
  • If the p-value is greater than the significance level, there is insufficient evidence to reject the null hypothesis
• Confidence interval provides a range of values within which the true population parameter is likely to fall, based on the sample data and the chosen confidence level (e.g., 95% confidence interval)

Types of Errors

Type I and Type II Errors

• Type I error (false positive) occurs when the null hypothesis is rejected even though it is actually true
  • The significance level ($\alpha$) represents the probability of making a Type I error
  • Example: Concluding a drug is effective when it actually has no effect
• Type II error (false negative) occurs when the null hypothesis is not rejected even though it is actually false
  • The probability of a Type II error is denoted by $\beta$
  • Example: Failing to detect a real difference between two treatment groups

Statistical Power

• Statistical power is the probability of correctly rejecting a false null hypothesis (i.e., detecting a true effect when it exists)
  • Power is calculated as $1 - \beta$, where $\beta$ is the probability of a Type II error
  • Factors that influence power include sample size, effect size, significance level, and the type of test used
• Increasing sample size, using a larger significance level, or focusing on larger effect sizes can increase statistical power

Test Statistics

Calculating Test Statistics

• Test statistic is a value calculated from the sample data that is used to make a decision about the null hypothesis
  • The choice of test statistic depends on the type of data and the research question (e.g., t-statistic, z-statistic, F-statistic)
  • Example: t-statistic is used to compare the means of two groups or to test the significance of a regression coefficient
• Degrees of freedom (df) represent the number of independent pieces of information used to calculate the test statistic
  • Degrees of freedom are determined by the sample size and the number of parameters being estimated
  • Example: In a two-sample t-test, df = $(n_1 + n_2 - 2)$, where $n_1$ and $n_2$ are the sample sizes of the two groups

Interpreting Test Statistics

• The test statistic is compared to a critical value determined by the degrees of freedom and the chosen significance level
  • If the test statistic exceeds the critical value, the null hypothesis is rejected
  • If the test statistic does not exceed the critical value, there is insufficient evidence to reject the null hypothesis
• The p-value associated with the test statistic indicates the probability of observing a test statistic as extreme or more extreme than the one calculated, assuming the null hypothesis is true