Inferential Statistics: Hypothesis Testing, Parametric & Non-Parametric Tests

While descriptive statistics summarize observed sample characteristics, inferential statistics enable researchers to draw mathematical inferences about unobserved population parameters from sample data. Inferential statistics determine whether observed experimental differences or correlations reflect genuine psychological phenomena or merely random sampling fluctuations (chance).

For the CUET PG Psychology examination (HUQP20), questions in this domain focus heavily on:

1. Sampling Distributions and the Central Limit Theorem (CLT): The Standard Error of the Mean ($SEM$) and confidence interval estimations.
2. Logic of Hypothesis Testing: Null ($H_0$) versus Alternative ($H_1$) hypotheses; Directional (One-Tailed) versus Non-Directional (Two-Tailed) testing.
3. Decision Errors and Statistical Power: The $2 \times 2$ decision matrix; Type I Error ($\alpha$), Type II Error ($\beta$), and Statistical Power ($1 - \beta$).
4. Parametric Significance Tests: Assumptions of parametric analysis; Student's $t$-test (independent and paired); Analysis of Variance (ANOVA, $F$-ratio, partitioning of variance, and post-hoc tests).
5. Correlational & Regression Analytics: Pearson Product-Moment Correlation ($r$), Coefficient of Determination ($r^2$), specialized correlational metrics (Biserial, Point-Biserial, Phi, Tetrachoric), Partial and Multiple correlation, and linear regression.
6. Non-Parametric (Distribution-Free) Tests: Chi-Square ($\chi^2$) test of goodness-of-fit and independence; Yates' correction; and the direct non-parametric counterparts to parametric tests (Mann-Whitney $U$, Wilcoxon Signed-Rank, Kruskal-Wallis $H$, and Friedman).

1. Foundations of Statistical Inference: Sampling Distributions & CLT

• Parameter vs. Statistic:
  • Parameter: A numerical characteristic describing an entire population (represented by Greek letters: Mean = $\mu$, Standard Deviation = $\sigma$, Correlation = $\rho$). Parameters are generally fixed, unknown values.
  • Statistic: A numerical value computed from a sample to estimate the corresponding population parameter (represented by Latin letters: Mean = $\bar{X}$, Standard Deviation = $s$, Correlation = $r$).
• Sampling Distribution of the Mean: A theoretical probability distribution of the means of an infinite number of random samples of size $N$ drawn from a specified population.
• The Central Limit Theorem (CLT): $\text{As sample size } N \text{ increases (typically } N \ge 30\text{), the sampling distribution of sample means}$ $\text{approaches a Gaussian normal distribution, regardless of the shape of the underlying population.}$
  • The mean of the sampling distribution equals the population mean: $\mu_{\bar{X}} = \mu$.
  • The standard deviation of the sampling distribution is called the Standard Error of the Mean ($SEM$ or $\sigma_{\bar{X}}$): $\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{N}} \qquad\qquad s_{\bar{X}} = \frac{s}{\sqrt{N}}$
• Confidence Intervals ($CI$) for the Population Mean ($\mu$):
  • Demonstrates the range within which the true population mean is estimated to lie with a specified degree of probability: $95\% \text{ Confidence Interval} = \bar{X} \pm (1.96 \times s_{\bar{X}})$ $99\% \text{ Confidence Interval} = \bar{X} \pm (2.58 \times s_{\bar{X}})$

2. Logic of Hypothesis Testing & Error Taxonomy

Hypothesis testing operates via proof by contradiction (reductio ad absurdum). Rather than proving the research hypothesis directly, science evaluates the probability of observing the sample data if the null hypothesis of "no difference" were true.

Directional vs. Non-Directional Tests

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• Non-Directional (Two-Tailed) Test: Hypothesizes that groups will differ, but does not specify the direction of the difference ($H_1: \mu_1 \neq \mu_2$). The rejection region ($\alpha$) is divided equally between both tails of the distribution ($2.5\%$ at the lower tail and $2.5\%$ at the upper tail for $\alpha = .05$; critical $z = \pm 1.96$).
• Directional (One-Tailed) Test: Hypothesizes an explicit direction of effect ($H_1: \mu_1 > \mu_2$ or $H_1: \mu_1 < \mu_2$). The entire critical rejection region ($\alpha = .05$) is concentrated in a single tail (critical $z = +1.645$ or $-1.645$). One-tailed tests possess greater statistical power to detect an effect in the predicted direction, but cannot detect an effect in the opposite direction.

The $2 \times 2$ Decision Matrix: Type I Error, Type II Error, and Statistical Power