What Is A T Test Explaining Core Concepts Applications And Interpretation

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A t-test is a cornerstone of statistical analysis, enabling researchers to evaluate differences between group means or paired observations with precision. By quantifying uncertainty through the t-statistic, this hypothesis-testing tool provides a rigorous framework for decision-making in fields ranging from clinical trials to social sciences. Its versatility—spanning one-sample, independent, and paired comparisons—makes it indispensable for validating experimental results, yet its effectiveness hinges on adherence to key assumptions like normality and variance homogeneity.

The method’s mathematical foundation, rooted in the t-distribution, balances simplicity with robustness, offering insights even with small sample sizes. Whether assessing drug efficacy, comparing pre- and post-intervention outcomes, or analyzing demographic disparities, t-tests deliver actionable conclusions when applied correctly. However, misapplication—such as ignoring non-normal data or overlooking unequal variances—can lead to flawed interpretations, underscoring the need for careful assumption checks and alternative approaches like Welch’s t-test or non-parametric alternatives.

what is a t test

Definition and Core Concept of a T-Test

The t-test is a fundamental statistical procedure used to determine whether the means of two or more groups differ significantly from one another or from a known value. Developed by William Sealy Gosset under the pseudonym "Student," the t-test relies on the t-distribution, which approximates the normal distribution but accounts for smaller sample sizes and greater variability. Its primary application lies in hypothesis testing, where it evaluates whether observed differences in sample means are statistically significant or attributable to random variation. The t-test is widely employed in fields such as medicine, psychology, economics, and engineering to make inferences about population parameters based on sample data.

The effectiveness of a t-test depends on adherence to key assumptions that ensure valid results. Violations of these assumptions can lead to incorrect conclusions, emphasizing the need for rigorous data assessment before application.

Fundamental Purpose and Role in Hypothesis Testing

The core objective of a t-test is to assess the null hypothesis (H₀) that no difference exists between group means (or between a sample mean and a population mean) against the alternative hypothesis (H₁) that a meaningful difference exists. This is achieved by calculating a test statistic (t-value), which quantifies the discrepancy between observed sample means relative to the variability within the data. The t-value is compared to a critical value from the t-distribution (determined by the degrees of freedom and significance level, α), or a p-value is computed to determine statistical significance.

The t-test is particularly valuable when:

  • Sample sizes are small (n < 30), where the normal distribution may not be a reliable approximation.
  • Population standard deviations are unknown, necessitating estimation from sample data.
  • The goal is to infer causality or effect size between groups under controlled conditions.

    • For example, a pharmaceutical company might use a t-test to compare the average blood pressure reduction between a treatment group and a placebo group, determining whether the observed effect is statistically significant.

    Core Assumptions of a T-Test

    The validity of a t-test hinges on three primary assumptions, each critical for ensuring accurate p-values and confidence intervals. Violations often necessitate alternative statistical methods, such as non-parametric tests or transformations.

    1. Normality of the Data
    The t-test assumes that the sampling distribution of the mean is approximately normal. For one-sample and independent t-tests, this implies that the population from which samples are drawn is normally distributed. For paired t-tests, the differences between paired observations should be normally distributed.

  • Implications: With small sample sizes (n < 30), non-normality can severely distort results. For larger samples (n ≥ 30), the Central Limit Theorem mitigates this concern due to the normal approximation of the sampling distribution.
  • Detection: Use Shapiro-Wilk test, Kolmogorov-Smirnov test, or Q-Q plots to assess normality. Skewed or kurtotic distributions may require transformations (e.g., log, square root) or non-parametric alternatives (e.g., Mann-Whitney U test).

    • 2. Homogeneity of Variance (Homoscedasticity)
    In independent t-tests, the assumption requires that the variances of the two groups being compared are equal. This is known as homogeneity of variance or homoscedasticity.

  • Implications: Unequal variances (heteroscedasticity) inflate Type I or Type II error rates. The t-test becomes conservative (less likely to detect true differences) when variances are unequal.
  • Detection: Use Levene’s test or F-test for equality of variances. If violated, consider Welch’s t-test (which does not assume equal variances) or unequal variance t-tests.

    • 3. Independence of Observations
    Observations within and between groups must be independent, meaning no relationship exists between any two data points beyond the grouping variable.

  • Implications: Violations occur with repeated measures, clustered data, or dependent samples (e.g., twins, matched pairs). This assumption is automatically satisfied in independent t-tests but critical in paired designs.
  • Detection: Check for autocorrelation, repeated measures, or pairing effects. Use intraclass correlation coefficients (ICC) or Durbin-Watson test for time-series data.

    • Comparison of T-Test Variants

    The selection of a t-test type depends on the research question, data structure, and experimental design. Below is a structured comparison of the three primary variants:
    Feature One-Sample T-Test Independent (Two-Sample) T-Test Paired T-Test
    Test Purpose Compares a single sample mean to a known population mean. Compares means between two independent groups (e.g., treatment vs. control). Compares means of the same subjects under two conditions (e.g., pre-test vs. post-test).
    Hypotheses
    H₀: μ = μ₀ (Sample mean equals population mean)

    H₁: μ ≠ μ₀ (or μ > μ₀, μ < μ₀ for one-tailed tests)
    H₀: μ₁ = μ₂ (Group means are equal)

    H₁: μ₁ ≠ μ₂ (or μ₁ > μ₂, μ₁ < μ₂)
    H₀: μ_d = 0 (Mean difference between pairs is zero)

    H₁: μ_d ≠ 0 (or μ_d > 0, μ_d < 0)
    Key Formula
    t = (x̄ - μ₀) / (s / √n)

    df = n - 1
    t = (x̄₁ - x̄₂) / √(s_p²(1/n₁ + 1/n₂))

    s_p² = pooled variance = [(n₁-1)s₁² + (n₂-1)s₂²] / (n₁ + n₂ - 2)

    df = n₁ + n₂ - 2 (for equal variances)

    Welch’s t-test (unequal variances):

    t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂)

    df ≈ (s₁²/n₁ + s₂²/n₂)² / [(s₁²/n₁)²/(n₁-1) + (s₂²/n₂)²/(n₂-1)]
    t = x̄_d / (s_d / √n)

    df = n - 1 (for paired differences)
    When to Use
    • Testing if a sample mean differs from a benchmark (e.g., "Is the average IQ of students in this school higher than the national average of 100?").
    • Quality control (e.g., "Does the mean weight of produced items deviate from the specified standard?").
    • Comparing two distinct groups (e.g., "Does a new drug reduce blood pressure more than a placebo?").
    • Experimental vs. control conditions with independent subjects.
    • Before-and-after measurements (e.g., "Did the training program improve employee performance?").
    • Matched pairs or repeated measures (e.g., "Is there a difference in reaction times between left and right hands?").

    Assessing Data Suitability for a T-Test

    Determining whether

    what is a t test - Ilustrasi 2

    Mathematical Foundations and Key Formulas of the T-Test

    The t-test is grounded in statistical theory that extends beyond the normal distribution, particularly when sample sizes are small or population parameters are unknown. Its mathematical formulation relies on estimating sample statistics and accounting for uncertainty through the t-distribution. The core of the t-test lies in the calculation of the t-statistic, which quantifies the discrepancy between observed data and a hypothesized value in units of estimated standard error. This section dissects the derivation of the t-statistic, compares its application across test types, and demonstrates its practical computation through a step-by-step example.

    Derivation of the T-Statistic Formula

    The t-statistic is derived from the standardized difference between a sample mean and a hypothesized population mean, adjusted for sample variability. The formula for the one-sample t-test is:

    t = (X̄ – μ₀) / (s / √n)

    Where:

  • X̄ (X-bar) represents the sample mean, calculated as:
    • X̄ = (ΣXᵢ) / n
    (Sum of all observations divided by the sample size n).
  • μ₀ (mu-zero) is the hypothesized population mean under the null hypothesis.
  • s is the sample standard deviation, computed as:
    • s = √[Σ(Xᵢ – X̄)² / (n – 1)]
    (Square root of the sum of squared deviations from the mean, divided by n – 1 degrees of freedom).
  • √n denotes the standard error of the mean (SEM), which scales the standard deviation by the square root of the sample size to account for reduced variability in larger samples.

    • The t-statistic follows a t-distribution with n – 1 degrees of freedom, which converges to the standard normal distribution (Z-distribution) as sample size increases. This adjustment is critical when sample sizes are small, as the t-distribution accounts for greater uncertainty in estimating the population standard deviation.

    Comparison of T-Test Formulas Across Test Types

    The structure of the t-statistic varies depending on the test type, reflecting differences in sample design and hypotheses. Below is a side-by-side comparison of the formulas for one-sample, independent-samples (two-sample), and paired-samples (dependent) t-tests, with annotations for each term.
    Component One-Sample T-Test Independent-Samples T-Test Paired-Samples T-Test
    Purpose Compare a sample mean to a known population mean. Compare means of two independent groups. Compare means of the same subjects under two conditions.
    Formula
    t = (X̄ – μ₀) / (s / √n)
    t = (X̄₁ – X̄₂) / √[(s₁²/n₁) + (s₂²/n₂)]
    t = (d̄) / (s_d / √n)
    Key Terms
    • X̄: Sample mean.
    • μ₀: Hypothesized population mean.
    • s: Sample standard deviation.
    • n: Sample size.
    • X̄₁, X̄₂: Means of Group 1 and Group 2.
    • s₁², s₂²: Variances of Group 1 and Group 2.
    • n₁, n₂: Sample sizes of Group 1 and Group 2.
    • Pooled variance (optional): Used when assuming equal variances (Welch’s t-test avoids pooling).
    • d̄: Mean of difference scores (X₁ – X₂ for paired observations).
    • s_d: Standard deviation of difference scores.
    • n: Number of pairs.
    Degrees of Freedom (df)
    df = n – 1
    df = n₁ + n₂ – 2 (for equal variances) or

    df = min(n₁ – 1, n₂ – 1) (Welch’s correction for unequal variances)
    df = n – 1
    Assumptions
    • Data normally distributed or sample size ≥ 30 (Central Limit Theorem).
    • Population standard deviation unknown.
    • Independent observations between groups.
    • Normality of data or large sample sizes.
    • Homogeneity of variances (for standard t-test; Welch’s test relaxes this).
    • Difference scores normally distributed.
    • Paired observations (e.g., pre-post measurements).

    Step-by-Step Calculation Example for a One-Sample T-Test

    Consider a hypothetical dataset where a researcher tests whether the mean score of a sample differs from a hypothesized population mean of 45. The sample consists of 30 observations with a sample mean (X̄) of 50 and a sample standard deviation (s) of 10. The null hypothesis (H₀) states that the population mean (μ) equals 45.

    Step 1: Identify Parameters

  • Sample mean (X̄) = 50
  • Hypothesized mean (μ₀) = 45
  • Sample standard deviation (s) = 10
  • Sample size (n) = 30

    • Step 2: Calculate the Standard Error of the Mean (SEM)
    The SEM accounts for the precision of the sample mean estimate:

    SEM = s / √n = 10 / √30 ≈ 10 / 5.477 ≈ 1.826
    Step 3: Compute the T-Statistic
    Substitute the values into the t-formula:
    t = (X̄ – μ₀) / SEM = (50 – 45) / 1.826 ≈ 5 / 1.826 ≈ 2.738
    Step 4: Determine Degrees of Freedom
    For a one-sample t-test:
    df = n – 1 = 30 – 1 = 29
    Step 5: Interpret the Result
    The calculated t-value of 2.738 with 29 degrees of freedom can be compared to critical t-values from a t-distribution table (e.g., ±2.045 for α = 0.05, two-tailed). Since 2.738 > 2.045, the null hypothesis is rejected, suggesting the sample mean significantly differs from the hypothesized population mean.

    Degrees of Freedom in T-Tests

    Degrees of freedom (df) represent the number of independent pieces of information available to estimate a parameter, directly influencing the shape of the t-distribution. In t-tests, df adjust for the loss of information due to sample constraints, particularly when estimating the population standard deviation from a sample.

    Types of T-Tests and Their Applications

    The t-test is a versatile statistical tool used to compare means across groups or against a known reference, but its application varies depending on the research design and data structure. Three primary types—one-sample, independent (two-sample), and paired (dependent) t-tests—address distinct hypotheses and real-world challenges. Understanding their distinctions ensures appropriate selection, which directly impacts the validity of conclusions. Below, these tests are examined through practical applications, comparative assumptions, and considerations for non-normal data to guide methodological decision-making.

    Real-World Applications of T-Test Types

    The selection of a t-test type depends on the experimental or observational structure of the data. Each variant serves distinct analytical purposes, from evaluating single-group deviations to comparing pre/post interventions or independent groups. Below are categorized scenarios demonstrating their utility, followed by a structured table of three practical examples per test type.

    One-sample t-test evaluates whether a sample mean significantly differs from a known population mean or hypothesized value. Applications include:

  • Quality control: Assessing whether a manufacturing batch’s average dimension meets industry standards.
  • Clinical trials: Determining if a drug’s effect in a patient sample deviates from a placebo baseline.
  • Academic performance: Comparing student test scores against a national average.

    • Independent (two-sample) t-test compares means between two distinct groups, such as treatment vs. control or demographic subgroups. Examples include:

  • Educational research: Evaluating the impact of a teaching method on test scores between experimental and control groups.
  • Social sciences: Investigating gender-based salary disparities in a corporate workforce.
  • Marketing: Testing consumer preference differences between two product variants in a survey.

    • Paired (dependent) t-test assesses mean differences within the same subjects or matched pairs, accounting for within-subject variability. Use cases include:

  • Medical studies: Measuring pre/post-treatment improvements in blood pressure for the same patients.
  • Psychology: Analyzing cognitive performance changes before and after a training program.
  • Economics: Comparing household income before and after a policy intervention.

    • Practical Examples of T-Tests in Research

    The following table summarizes three real-world applications for each t-test type, detailing the research question, data structure, statistical method, and interpretation of results. These examples illustrate how t-tests address diverse hypotheses while adhering to experimental constraints.
    Research QuestionData StructureTest SelectedInterpretation of Results
    One-Sample T-Test
    Does a new fertilizer increase crop yield above the industry average of 50 kg/ha?Single sample (n=40 fields), continuous yield data.One-sample t-test (μ₀ = 50 kg/ha).If p < 0.05, reject H₀; conclude the fertilizer significantly exceeds the benchmark.
    Is the average reaction time of gamers (sample mean = 220 ms) slower than the general population (μ₀ = 200 ms)?Single sample (n=100 gamers), reaction time data.One-sample t-test (μ₀ = 200 ms).Significant p-value indicates slower reaction times in gamers compared to the population norm.
    Does the mean IQ of a gifted education program (sample mean = 130) differ from the national average (μ₀ = 100)?Single sample (n=50 students), IQ scores.One-sample t-test (μ₀ = 100).A non-significant result suggests the program’s mean IQ aligns with the national average.
    Independent T-Test
    Does a new pain medication reduce recovery time compared to a placebo in patients?Two groups (treatment: n=60, placebo: n=60), recovery time (days).Independent t-test (equal variances assumed).Significant difference (p < 0.05) implies the drug accelerates recovery.
    Are there differences in math test scores between urban and rural schools?Two groups (urban: n=150, rural: n=120), test scores.Independent t-test (Welch’s if variances unequal).If p < 0.05, conclude urban/rural disparities exist in academic performance.
    Does employee satisfaction differ between remote and office-based workers?Two groups (remote: n=80, office: n=70), satisfaction scores (1–10).Independent t-test (check Levene’s test for homogeneity).Non-significant result may indicate no difference in satisfaction between work models.
    Paired T-Test
    Does a weight-loss program reduce participants’ BMI after 12 weeks?Paired data (pre/post BMI for n=30 individuals).Paired t-test.Significant reduction (p < 0.05) validates the program’s efficacy in lowering BMI.
    Are there improvements in memory recall scores after cognitive training?Paired data (pre/post scores for n=45 subjects).Paired t-test.A significant p-value confirms training enhances memory recall.
    Does a new tire tread design improve fuel efficiency compared to the standard model?Paired data (fuel efficiency for n=20 vehicles, tested with both treads).Paired t-test.Positive mean difference with p < 0.05 indicates superior performance of the new tread.

    Assumptions and Limitations: Student’s vs. Welch’s T-Test

    The Student’s t-test and Welch’s t-test share a common objective—comparing means—but differ in their treatment of variance assumptions. Student’s t-test assumes homogeneity of variances (equal population variances) and uses a pooled variance estimator, while Welch’s t-test relaxes this assumption by employing separate variance estimators for each group. This distinction is critical for ensuring valid inferences, particularly when group sizes or variances differ.

    Key Assumptions and Trade-offs:

  • Student’s t-test:
  • Assumptions: Normality of data, homogeneity of variances (verified via Levene’s or Bartlett’s test), independent samples.
  • Limitations: Sensitive to unequal variances; inflated Type I error risk if violated. Less robust with small, unequal sample sizes.
  • Application: Preferred when variances are homogeneous and sample sizes are balanced.

    • - Welch’s t-test:

  • Assumptions: Normality of data, no requirement for equal variances; independent samples.
  • Advantages: More robust to unequal variances and sample sizes; reduces Type I error when variances differ.
  • Limitations: Slightly less statistical power when variances are truly equal; degrees of freedom adjusted downward.
  • Application: Default choice when Levene’s test rejects homogeneity or sample sizes are unequal.

    • When to Use Welch’s T-Test:

  • Group variances are significantly unequal (p < 0.05 in Levene’s test).
  • Sample sizes differ substantially (e.g., n₁ = 20, n₂ = 100).
  • Data distribution is normal or approximately normal (non-normality may still require non-parametric tests).

    • Formula Adjustment in Welch’s Test:
    The degrees of freedom (df) are calculated as:

    \[
    df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{(s_1^2/n_1)^2}{n_1-1} + \frac{(s_2^2/n_2)^2}{n_2-1}}
    \]
    where \(s_1^2\) and \(s_2^2\) are the sample variances, and \(n_1\) and \(n_2\) are the sample sizes. This adjustment accounts for unequal variances, providing a more accurate p-value.

    Handling Non-Normal Data in T-Tests

    T-tests rely on the assumption of normally distributed data, particularly for small samples (n < 30). Violations of this assumption can lead to inflated Type I or II errors. When normality is compromised, researchers must consider data transformations, robust alternatives, or non-parametric tests, each with distinct trade-offs in terms of power and interpretability.

    Approaches to Address Non-Normality:
    1. Data Transformations:

  • Log transformation: Useful for right-skewed data (e.g., income, reaction times). Stabilizes variance and normalizes distributions.
  • Square root transformation: Applied to count data or moderately skewed distributions.
  • Box-Cox transformation: Generalized method to find optimal power transformation (λ) for normality.
  • Trade-offs: May distort interpretability (e.g., log-transformed means are not directly meaningful in original units); requires back-transformation for

    • what is a t test - Ilustrasi 3

    Interpreting T-Test Results and Effect Sizes

    The interpretation of t-test results extends beyond mere statistical significance to include effect sizes, confidence intervals, and practical implications. Researchers must evaluate whether observed differences are meaningful in real-world contexts, not just mathematically significant. This section provides structured templates for reporting t-test findings in APA style, clarifies the distinction between statistical and practical significance, and demonstrates effect size calculation using Cohen’s d. Additionally, a comparative analysis of p-values and confidence intervals highlights their roles in decision-making, including potential biases.

    Template for Reporting T-Test Results in APA Style

    APA formatting requires precise reporting of t-test statistics, effect sizes, and confidence intervals to ensure reproducibility and clarity. Below is a template with placeholders for hypothetical results, structured for a two-sample independent t-test (adjust for paired or one-sample variants as needed).

    Example for Independent Samples T-Test:
    > An independent-samples t-test was conducted to compare the mean scores of Group A (M = X, SD = Y, n = Z) and Group B (M = X, SD = Y, n = Z). The results indicated a significant difference in [variable] between the groups, t(df = N) = t-value, p = .XXX. The 95% confidence interval for the difference between means ranged from [lower bound] to [upper bound]. Cohen’s d was calculated as d-value, indicating a [small/medium/large] effect size.

    Key Placeholders:

  • Replace X, Y, Z, N, t-value, .XXX, and bounds with actual values.
  • For paired t-tests, specify "paired-samples t-test" and adjust df to n–1.
  • Include effect size metrics (e.g., d or η²) where applicable.

    • Example for One-Sample T-Test:
    > A one-sample t-test was performed to evaluate whether the mean score of Sample (M = X, SD = Y, n = Z) differed significantly from the population mean of X₀. The analysis yielded t(df = N) = t-value, p = .XXX, with a 95% CI ranging from [lower] to [upper]. The effect size, measured as d = value, suggests [interpretation].

    Statistical Significance vs. Practical Significance

    Statistical significance, as indicated by the p-value, assesses whether an observed effect could have occurred by chance under the null hypothesis. Practical significance, however, evaluates whether the effect is meaningful in applied contexts. These concepts are often conflated, leading to misinterpretations of research findings.
    The p-value answers the question: "Is the observed difference unlikely under the null hypothesis?" It does not quantify the size or importance of the effect. In contrast, effect size (e.g., Cohen’s d, η²) measures the magnitude of the difference relative to variability, providing insight into practical relevance. A low p-value (p < .05) may accompany a trivial effect size, while a high p-value may mask a meaningful difference in large samples. Both metrics are essential for rigorous inference.
    Common Effect Size Metrics for T-Tests:
    1. Cohen’s d (Standardized mean difference):
  • Interpretation: Small (0.2), Medium (0.5), Large (0.8).
  • Formula: d = (M₁ – M₂) / pooled SD.
  • Used for independent or paired t-tests.

    • 2. Hedges’ g (Bias-corrected d):

  • Preferred for small sample sizes (n < 20) due to reduced bias.

    • 3. Partial η² (Variance explained):

  • Formula: η² = t² / (t² + df).
  • Interpretation: Small (0.01), Medium (0.06), Large (0.14).

    • 4. Pearson’s r (Correlation-based effect size):

  • Derived from t for independent samples: r = t / √(t² + df).

    • Calculating and Interpreting Cohen’s d for a Two-Sample T-Test

    Cohen’s d standardizes the difference between two means by the pooled standard deviation, enabling cross-study comparisons. Below is a step-by-step calculation using a hypothetical dataset comparing exam scores between two teaching methods.

    Hypothetical Data:

  • Method A (n = 30): Mean (M₁) = 78, SD (s₁) = 8.
  • Method B (n = 30): Mean (M₂) = 82, SD (s₂) = 7.

    • Step 1: Compute the Pooled Standard Deviation
    The pooled SD accounts for variability in both groups, weighted by sample size:
    \[
    s_{pooled} = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}
    \]
    \[
    s_{pooled} = \sqrt{\frac{(29)(8^2) + (29)(7^2)}{30 + 30 - 2}} = \sqrt{\frac{1856 + 1421}{58}} = \sqrt{53.48} \approx 7.31
    \]

    Step 2: Calculate Cohen’s d \[
    d = \frac{M_1 - M_2}{s_{pooled}} = \frac{78 - 82}{7.31} = \frac{-4}{7.31} \approx -0.55
    \]
    Interpretation: The negative value indicates Method B outperformed Method A by 0.55 SDs, corresponding to a medium effect size (|d| = 0.5).

    Thresholds for Cohen’s d:

    Effect Size (d)Interpretation
    0.2Small
    0.5Medium
    0.8Large
    Note: For paired t-tests, use the mean difference and standard deviation of differences:
    \[
    d = \frac{M_{diff}}{s_{diff}}
    \]

    Comparison of P-Values and Confidence Intervals in Decision-Making

    While p-values and confidence intervals (CIs) both inform hypothesis testing, they provide distinct perspectives. The table below contrasts their use cases, advantages, and potential pitfalls in research decisions.
    The t-test remains a vital instrument in statistical inference, bridging theoretical rigor with practical applicability across disciplines. By mastering its core concepts—from formulaic calculations to nuanced result interpretation—analysts can confidently distinguish meaningful effects from random variation. Whether evaluating a single group’s deviation from a standard, comparing two independent samples, or analyzing paired differences, the t-test’s structured approach ensures reliability when assumptions are met. Yet, its limitations—particularly with non-normal distributions—demand adaptability, reinforcing the importance of complementary methods and effect size metrics like Cohen’s d to contextualize findings. Ultimately, the t-test exemplifies how statistical tools, when wielded thoughtfully, transform raw data into evidence-based insights.

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    Scenario P-Value Approach Confidence Interval Approach Potential Pitfalls
    Small Sample Size (n < 30) Relies on t-distribution; sensitive to non-normality. Wider intervals reflect greater uncertainty; less precise estimates.
    • P-values may overstate significance due to t-distribution assumptions.
    • CIs may exclude zero even with trivial effects if sample size is small.
    Large Sample Size (n > 100) Minor differences become "significant" (p < .05) due to high power. Narrow intervals provide precise estimates of effect magnitude.
    • Statistical significance ≠ practical relevance (e.g., p < .05 for d = 0.01).
    • CIs may be misleadingly narrow, suggesting precision where none exists.
    Non-Normal Data Parametric t-tests assume normality; violations inflate Type I error. Robust to non-normality if sample size is large (CLT applies).
    • P-values may be inaccurate for skewed data.
    • CIs may undercover true uncertainty if normality is violated.
    Equivalence Testing Cannot directly assess equivalence; only tests for difference. Explicitly evaluates whether the effect falls within a "practical" range (e.g., ±5%).