The Breakdown
Important
- Critical Z-scores – These are specific, set Z-values (e.g., +/- 1.65, +/- 1.96, +/- 2.33, +/- 2.58) that define the boundaries of the critical rejection region on the standard normal curve, varying based on the alpha level and whether the hypothesis is one-tailed or two-tailed. They are essential for determining whether to reject the null hypothesis.
- Four Pairings of Significance and Effect Size – Understanding these four combinations (Reject H0/Small d = Type I error; Reject H0/Moderate-Large d = Meaningful; Fail to Reject H0/Small d = Abandon; Fail to Reject H0/Moderate-Large d = Type II error) is crucial for interpreting research results and determining the likelihood of making a Type I or Type II error. This relationship is a cornerstone for the entire course.
- Steps of Hypothesis Testing – The process of hypothesis testing (stating hypotheses, determining alpha and critical Z, computing statistics, comparing ZM to critical Z, calculating effect size, and reporting results) is a structured, sequential process that must be followed precisely, with marks allocated for each step in assignments and midterms.
- Effect Sizes – These are measures that are independent of sample size and quantify how meaningful research findings are. It is crucial to report effect sizes alongside statistical significance to provide a comprehensive understanding of the results and to indicate the probability of committing a Type I or Type II error.
Core Concepts
- Probability: The likelihood of something occurring beyond chance level, defined as the proportion or percentage of times an event would occur if chances for occurrence were infinite. It is not about certainties or guarantees, but rather long-run patterns, typically based on speculation, intuition, previous experience, and knowledge of theoretical outcomes.
- Random Sampling: A crucial criterion for confident research results, where every person in the population has an equal chance of being selected, and there is a constant probability of being selected (sampling with replacement, though this is often overlooked with large populations due to minimal impact).
- Probability and the Standard Normal Curve: Once a Z-score for an individual score or sample mean is calculated, it can be located on the standard normal curve to determine the probability associated with that score or mean. The curve’s area totals 100%, allowing proportions to be equated to probabilities.
- Normal Curve Table (Appendix B1): Used to find the proportion (probability) corresponding to a Z-score by looking up the Z-score under either the “body” column (for the larger portion of the curve, typically below a positive Z or above a negative Z) or the “tail” column (for the smaller portion, typically above a positive Z or below a negative Z).
- Sampling Distribution of the Means: When multiple random samples of the same size are drawn from a population, each yielding a slightly different mean, plotting these sample means will form a normal curve, especially when there are more than 30 samples.
- Central Limit Theorem: States that the mean of all sample means (μM) equals the mean of the parent population (μ). It also asserts that the larger the sample size, the more the sampling distribution of means will approximate a normal curve (typically when n > 30), regardless of the population’s original shape.
- Standard Error of the Mean (σM): The standard deviation of the sampling distribution of means, calculated by dividing the population standard deviation (σ) by the square root of the sample size (√n). It indicates the variability of sample means around the population mean and is smaller with larger sample sizes.
- Z-score for the Mean (ZM): A transformation that converts a sample mean into a Z-score, allowing for the determination of the probability of obtaining that specific sample mean from a known population. It is calculated as (Sample Mean – Population Mean) / Standard Error of the Mean.
- Hypothesis Testing: A research protocol that begins with making predictions about an intervention’s expected outcome before data collection. It involves testing whether a sample drawn from a population is sufficiently different from it or representative of it, often using Z-scores associated with sample means.
- Research Hypothesis (H1) / Alternative Hypothesis: The statement that represents the expected outcome or the researcher’s prediction, suggesting an effect or difference due to the intervention.
- Null Hypothesis (H0): The statement that is the opposite of the research prediction, typically asserting that the treatment or intervention will have no effect, or that there is no difference between the sample and the population.
- Directional (One-tailed) Hypothesis: Predicts a specific direction for the effect (e.g., “greater than” or “lower than”). All the alpha (critical region) is placed on one side of the distribution.
- Non-directional (Two-tailed) Hypothesis: Predicts that there will be a difference, but does not specify the direction (e.g., “different from,” “not the same”). The alpha is split evenly between both tails of the distribution.
- Alpha (α) Level: The chosen proportion (e.g., 0.05 or 0.01) of the standard normal curve that defines the critical rejection region. It represents the researcher’s acceptable probability of making a Type I error (concluding an effect exists when it doesn’t).
- Critical Zones/Regions: The extreme areas under the standard normal curve (defined by the alpha level and critical Z-scores) where, if a ZM score falls, it suggests a very low probability of occurring by chance, leading to the rejection of the null hypothesis and concluding a significant difference from the population.
- Conclusions of Hypothesis Testing: Either Fail to reject the null hypothesis (if the ZM does not fall into the critical region, meaning the sample is not significantly different from the population) or Reject the null hypothesis (if the ZM falls into the critical region, meaning the sample is significantly different from the population, implying the treatment was effective).
- Type I Error (α): Occurs when the researcher rejects the null hypothesis when it was actually true (a false claim of an effect). The alpha level set for research directly corresponds to the probability of making this error. This error can be serious, especially in medical research.
- Type II Error (β): Occurs when the researcher fails to reject the null hypothesis when it was actually false (a missed claim or failure of detection). This means a real effect was not detected. Beta is the opposite of alpha (e.g., if α=0.05, β=0.95), and this error is generally considered less serious as research can be redone.
- Power of a Study: The ability of a study to find a significant result if one truly exists. It is influenced by sample size (larger = more power), the actual size of the treatment effect, the chosen alpha level (larger alpha = more power), and the choice of a directional hypothesis (more power than a two-tailed test).
- Cohen’s d: The specific effect size measure used for the Z-test. It quantifies how much the independent variable (treatment) changed the mean of the dependent variable, expressed in standard deviation units. Cohen’s d values of 0.2, 0.5, and 0.8 are considered small, medium, and large effects, respectively.
- Reporting Results: The final step in hypothesis testing, which involves summarizing the findings by stating the conclusion regarding the hypothesis, including the sample mean, ZM score, probability (p < .05 or ns), and the interpretation of the effect size to convey the meaningfulness of the finding.
- Assumptions of the Z-test: Criteria that must be met before conducting the test, including random sampling, independent participants (no repeated measures or related individuals), and the assumption that the sample data is normally distributed.
Theories and Frameworks
- Standard Normal Curve: A theoretical bell-shaped distribution with a mean of 0 and a standard deviation of 1, used as a universal framework for understanding and comparing probabilities associated with Z-scores.
- Central Limit Theorem: A fundamental theorem in statistics that explains how the distribution of sample means, drawn from a population, will approximate a normal distribution as the sample size increases, and how the mean of these sample means will equal the population mean.
Notable Individuals
- Cohen: Developed Cohen’s d, a widely used measure of effect size to quantify the magnitude of a treatment effect.