10. Inferential Data Analysis
As the researcher draws scientific conclusions from his study using only a sample instead of the whole population, he can justify his conclusion with the help of statistical inference tools. The principal concepts involved in statistical inference are the theory of estimation and hypothesis testing.
Theory of Estimation
Point Estimation: A single value is used to provide the best estimate of the parameter of interest.
Interval Estimation: Interval estimates show the estimate of the parameter and also give an idea of the confidence that the researcher has in that estimate. This leads us to the consideration of confidence intervals.
Confidence Interval (CI): A confidence interval estimate of a parameter consists of an interval, along with a probability that the interval contains the unknown parameter. The level of confidence is denoted (1 - α)*100%.
The narrower the width of the confidence interval, the lower is the error of the point estimate it contains. The sample size, sample variance, and the level of confidence all affect the width. Reportings typically require 95% CIs.
Hypothesis Testing
The basic concept used in hypothesis testing is that it is far easier to show that something is false than to prove that it is true. Mutually exclusive hypotheses are created: Null Hypothesis (H0) and Alternative Hypothesis (H1). p-values determine if changes are statistically significant under alpha parameters.
Statistical Tests Parameters Matrix
| Variable Type | Parameters Tested | Variables Count | Criteria / Assumption | Appropriate Test |
|---|---|---|---|---|
| Ratio Variable | Mean | One | >30 (Large Sample) | Z-test |
| Ratio Variable | Mean | Two | >30 (Large Sample) | Z-test for Two Means |
| Ratio Variable | Mean | One | <30 (Small Sample) | t-test |
| Ratio Variable | Mean | Two | <30 (Independent) | Independent sample t-test |
| Ratio Variable | Mean (Same Subject) | Two | <30 (Paired) | Paired sample t-test |
| Ratio Variable | Proportion | One | - | Binomial |
| Ratio Variable | Proportion | Two | >30 (Large) | Z-test for Proportions |
| Ratio Variable | Mean | More than two | >30 | ANOVA |
| Ratio Variable | Mean (Same Subject) | More than two | >30 | Repeated measures ANOVA |
| Nominal/Categorical | Association | Two or more | Expected Frequency ≥ 5 | Chi-square |
| Ratio Variable | Mean | Two | Normality Assumption Violated | Mann-Whitney test |
| Ratio Variable | Mean (Same Subject) | Two | Normality Assumption Violated | Wilcoxon signed rank test |
| Ratio Variable | Mean | More than Two | Normality Assumption Violated | Kruskal Wallis test |