t-Tests and Inference: One-Sample and Independent-Samples (Comprehensive Notes)

One-Sample t-Test: concepts and interpretation

• Purpose: tests whether the mean of a single sample differs from a known population mean μ.
• Key language shift: raw scores (X scores) vs. t-scores. The t-score is a standardized value expressed in units of standard error.
• Raw score example from transcript:
  • Population mean μ = 40
  • Sample mean X̄ = 42
  • Sample size n = 36
  • Reported t ≈ 2.03 (positive because X̄ > μ)
• Formula for the one-sample t-statistic:
  $t = \frac{\overline{X} - \mu}{\dfrac{s}{\sqrt{n}}}$
  where $s$ is the sample standard deviation and $SE = \dfrac{s}{\sqrt{n}}$ is the standard error of the mean.
• Interpretation of the t-score:
  • Positive t indicates the sample mean is above the population mean.
  • Negative t indicates the sample mean is below the population mean.
  • The population mean has t = 0 (zero standard errors away from itself).
• Degrees of freedom (df):
  $df = n - 1$
  Rationale: use the sample SD to estimate the population SD; subtract 1 to reserve degrees of freedom for variability in estimation.
• P-value and decision rule:
  • For a two-tailed test, the p-value is the proportion of the sampling distribution as extreme or more extreme than the observed t in either tail:
    $p = P\left(|T| \ge |t<em>{obs}| \mid H</em>0\right)$
  • Example from transcript: observed t = 2.03, with reported p ≈ 0.042 (two-tailed).
  • Alpha (significance level) choice: commonly $\alpha = 0.05$; sometimes $\alpha = 0.01$ (e.g., FDA drug trials).
  • Decision rule:
  • If p < \alpha, reject the null hypothesis (conclude a difference exists).
  • If $p \ge \alpha$, retain (fail to reject) the null hypothesis (no evidence of a difference).
• Reporting results in APA format:
  • Statistical symbols in italics: T and P (and the test statistic type).
  • Include degrees of freedom after the T (e.g., t(35)).
  • Report the test statistic and p-value: e.g., $t(35) = 2.03,\quad p = .042$
• Conceptual interpretation recap:
  • The t-test assesses whether the sample mean is a significantly different value from the population mean.
  • “Retaining the null” means the two things are the same or not significantly different.
  • “Rejecting the null” means the two things are significantly different.
• Visualization intuition:
  • The sampling distribution of the mean under H0 is centered at μ with standard error SE.
  • The observed t-score marks a cut-off point; areas beyond ±t_obs form the critical region for a given α.
  • The p-value is the total area in the tails beyond the observed t in both directions for a two-tailed test.
• Example recap from transcript:
  • Population mean μ = 40; sample mean X̄ = 42; n = 36; t = 2.03; p ≈ 0.042; α = 0.05 → reject H0; conclude there is a difference between sample and population means.
• Connection to prior ideas:
  • t scores express how far the sample mean is from the population mean in units of SE, linking to the concept of standard error and the sampling distribution.
  • The null hypothesis and its rejection are evaluated via a distribution of possible sample means under H0.

The Language of X scores vs. t scores: practical notes

• X scores: raw measurements (e.g., extraversion score on a scale).
• T scores: transformed into a standardized metric using the standard error; tell you how many SEs the sample mean is from the population mean.
• Transformation intuition: converting to t scores allows comparison across studies with different scales, because you’re measuring deviation in units of SE rather than raw units.
• Example interpretation:
  • If a test on extraversion scale has population mean μ = 40 and the sample mean is X̄ = 42 with SE = 1, then $t = (42 - 40)/1 = 2.0$, indicating the sample mean sits 2 standard errors above μ.
• Practical steps you’ll perform (in practice, often by computer):
  • Compute t, df, and p-value; decide to reject or retain H0; report results.
  • In lab/PSPP outputs, the computer supplies t, df, and p-value; you then format the result for publication.

Two-tail vs. one-tail tests and the p-value interpretation

• Two-tailed test: tests whether the two means differ (either direction).
  • Alpha split across tails: $\alpha / 2$ in each tail (e.g., 0.025 per tail for $\alpha = 0.05$).
  • The p-value reflects both tails: the probability of being as extreme or more extreme in either direction.
• One-tailed test: tests for difference in a specified direction (not used in the transcript but often taught).
• In this lecture, emphasis is on two-tailed interpretation and the symmetrical t distribution.

Independent-samples t-test: concept and workflow

• Purpose: compare the means of two independent groups on a numeric dependent variable.
• Key structure:
  • Independent variable: nominal with exactly two levels (e.g., work vs not work; meat-eaters vs vegetarians).
  • Dependent variable: numeric (interval or ratio scale).
• Example variants discussed:
  • People who work vs people who don’t: numeric outcome could be hours of productivity at work or another measurable metric.
  • Meat-eaters vs vegetarians: numeric outcome could be muscle mass or another measurable trait.
• Two-group design intuition:
  • The independent variable partitions the sample into two groups.
  • The t-test assesses whether the two group means differ on the numeric outcome.
• Classic example from Kitona (1940) on memorization vs understanding:
  • Two learning methods: memorization vs learning by understanding.
  • Benefit observed for understanding method in a memorization task (the matchstick problem).
  • Design: there were multiple problems (e.g., 12 problems like the matchstick task) to compare learning methods.
  • Finding: learning by understanding yielded better outcomes than memorization, illustrating a difference between two learning approaches.
• Reporting and interpretation basics parallel the one-sample case but use a pooled or alternative variance estimate depending on equal-variance assumptions (noted in practice with more advanced variants).
• Example outcomes and reporting follow APA conventions similar to the one-sample case but with two sample means and the corresponding df.

Formulas and key statistical details to memorize

• One-sample t-test statistic: $t = \frac{\overline{X} - \mu}{\dfrac{s}{\sqrt{n}}}$
  • Degrees of freedom: $df = n - 1$
  • Standard error: $SE = \dfrac{s}{\sqrt{n}}$
• Two-sample independent t-test statistic (common version with pooled variance):
  • Pooled variance:
    $s<em>p^2 = \frac{(n</em>1 - 1)s<em>1^2 + (n</em>2 - 1)s<em>2^2}{n</em>1 + n_2 - 2}$
  • Test statistic:
    $t = \frac{\overline{X}<em>1 - \overline{X}</em>2}{\sqrt{ s<em>p^2 \left( \frac{1}{n</em>1} + \frac{1}{n_2} \right) }}$
  • Degrees of freedom:
    $df = n<em>1 + n</em>2 - 2$
• P-value for two-tailed tests:
  $p = P\left( |T| \ge |t_{obs}| \right)$
• Critical regions (for a two-tailed test):
  • Cutoffs at $\pm t_{\alpha/2, df}$, where the p-value is the area beyond these cutoffs in both tails.
• Reporting conventions (APA style):
  • Use italics for statistical symbols: T, P; include the degrees of freedom after T (e.g., t(35)).
  • Typical reporting format: $t(df) = t<em>{obs},\quad p = p</em>{value}$
  • Example: $t(35) = 2.03,\quad p = .042$

Practical interpretation and exam-ready tips

• Always state the null hypothesis clearly: there is no difference between the two means.
• Interpret t-score directionally: positive t indicates the sample mean is greater than the comparison mean; negative t indicates it is smaller.
• Decide on alpha ahead of time (e.g., 0.05); understand how two-tailed tests split alpha across both tails.
• Remember the logic: small p-value means the observed extreme difference would be unlikely if the null were true; thus reject H0.
• Note: The relationship between t, p, and alpha is a decision rule, not a claim of practical importance by itself; consider effect size and confidence intervals in addition to p-values.

Connections to broader concepts and real-world relevance

• The t-distribution arises due to estimating the population SD from the sample, which introduces extra uncertainty captured by df.
• The standard error reflects sampling variability; larger n reduces SE and makes it easier to detect smaller differences.
• APA formatting considerations reflect discipline-wide reporting standards and clarity in conveying results to readers.
• The two-sample t-test framework applies to a wide range of comparisons in psychology and social sciences (e.g., group differences in behavior, performance, or physiology).
• Ethical and practical implications: choosing appropriate alpha levels balances the risk of false positives and false negatives; stricter alpha (e.g., 0.01) reduces false positives but requires stronger evidence to declare a difference.

Quick recap of key terms

• Raw score (X-score): the original measurement value.
• t-score: standardized score in units of the standard error.
• Standard error (SE): the standard deviation of the sampling distribution of the mean, $SE = \dfrac{s}{\sqrt{n}}$.
• Degrees of freedom (df): the sample size parameter that influences the shape of the t-distribution.
• p-value: probability, under H0, of obtaining a test statistic as extreme or more extreme than observed.
• Alpha (\alpha): pre-specified threshold for deciding whether to reject H0.
• Critical region: values of the test statistic that lead to rejection of H0 for a given \alpha and df.
• APA format conventions: italicize statistical symbols, report t and p with df, etc.