Psy 350 study

Confidence interval

range of plausible population values.

Type I error (α):

reject a true H₀ (false positive)

Type II error (β):

fail to reject a false H₀ (false negative).

Variance (s²):

average squared spread around the mean

Standard deviation (s)

typical spread of scores, √variance.

Standard error (SE)

estimated variability of a sample statistic across samples.

Homogeneity of variance

populations have equal/similar variances.

Effect size

magnitude of a relationship/difference.

Variability

how spread out scores are

Power (1 − β)

): probability of detecting a real effect.

Random sampling

randomly choosing people from a population.

Random assignment

randomly placing participants into conditions.

ANOVA:

analysis comparing group means using variance.

Omnibus test:

overall test for any group difference

 Post hoc tests

follow-up group comparisons

Factor: independent variable (IV). Level

condition/category of a factor

Cell

unique combination of factor levels

MS

mean square = SS/df.

Main effect

effect of one factor averaged across another.

Interaction

effect of one IV depends on another IV

Factorial design:

research design with 2+ IVs

When are t statistics used in lieu of z scores?

Use t instead of z when the population standard deviation (σ) is unknown and is estimated using the sample SD (s).

Why do we need a different measure of standard error for t tests than z tests?

A z-test uses known σ; a t-test uses sample s to estimate σ, so its standard error is estimated (s/√n).

How does sample size influence the degree to which a t-statistic approximates a z-score?

Larger samples → larger df → t distribution more closely resembles the normal (z) distribution. Small samples produce heavier tails.

What are degrees of freedom? What abbreviation is used to represent degrees of freedom?

Degrees of freedom (df) = number of values free to vary after estimating parameters. Single-sample and paired t: df = n − 1; pooled independent t: df = n₁ + n₂ − 2.

State the null and alternative hypotheses for each t statistic.

Single-sample: H₀: μ = μ₀; H₁: μ ≠ μ₀. Independent-samples: H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂. Related-samples: H₀: μD = 0; H₁: μD ≠ 0. For directional tests, use > or < instead of ≠.

What are the guidelines for determining the size of Cohen’s d  (e.g., small, medium, large)?

Cohen’s |d|: 0.20 = small, 0.50 = medium, 0.80 or more = large.

How do interval estimates differ from point estimates?

Point estimate: one value estimating a parameter (e.g., M). Interval estimate: range of plausible values (e.g., 95% CI), showing uncertainty

What is the purpose of the independent-measures t- test?

To decide whether two independent groups' means differ beyond what sampling error would explain.

Describe the procedure to compute the independent-measures t test.

1) State H₀/H₁ and α (usually .05).

2) Obtain each group's mean, SD, n, and df.

3) For equal-variance t, calculate pooled variance s²p = (SS₁ + SS₂)/(df₁ + df₂).

4) SEdiff = √[s²p(1/n₁ + 1/n₂)].

5) t = (M₁ − M₂)/SEdiff; df = n₁+n₂−2.

6) Compare p with α and report results. (Welch’s test uses a different SE/df when equal variances are not assumed.)

What assumptions must be met for the independent-measures t test to yield a correct decision?

Independent observations; approximately normal population distributions; homogeneity of variance for the pooled/equal-variance version. If variances differ, use Welch’s t-test.

In which research situation is it appropriate to use a single-sample t test?

One sample mean compared to a known or hypothesized population mean, with population SD unknown (e.g., class mean vs. μ = 3).

In which research situation is it appropriate to use an independent-measures t test?

Two separate, unrelated groups compared on a quantitative outcome (e.g., feedback group A vs. group B).

In what two research situations is it appropriate to use a related-samples t test?

(1) Same participants tested twice/in two conditions (repeated measures). (2) Two separate groups matched into pairs based on a relevant trait.

How does a within-subjects design differ from a between-subjects design?

Within-subjects: same people complete all conditions. Between-subjects: different people take part in each condition.

What are the advantages of a repeated-measures design?

Uses fewer participants; controls individual differences; reduces error/variability and often improves power.

What are the disadvantages of a repeated-measures design?

Order/carryover effects (practice, fatigue); possible attrition or dropout between measurements.

What is the difference between data formatted the long way vs. the short way?

Long format: each participant-condition measurement is a separate row, with ID and condition columns. Short/wide format: one row per participant with separate columns for conditions.

When looking at the t-statistic as a ratio, how can we describe the numerator?

Numerator = observed difference (sample mean − hypothesized mean, difference between means, or mean paired difference); the apparent effect.

When looking at the t-statistic as a ratio, how can we describe the denominator?

Denominator = estimated standard error; the expected random/sampling variability in that difference.

How do you interpret an R output for each of the three types of t-tests? What values should be reported in APA format?

For all three, find t, df, p, 95% CI, and mean(s). One-sample: compare M to μ₀. Independent: compare group means. Paired: compare matched/within-person means or mean difference. Report descriptives (M, SD), t(df), p, an effect size (d), and CI when available. Example: t(28) = 2.31, p = .028, d = 0.42.

What criteria must be met in order to reject a null hypothesis / claim significance for a t-test?

Reject H₀ if p < α (usually .05), or if |t| exceeds the appropriate critical t. Otherwise, fail to reject H₀. A confidence interval for the tested difference that excludes 0 also indicates significance at the corresponding two-sided α.

How do you interpret the direction of t-test results?

Examine the group means and subtraction order. Positive t means first mean > second (given M₁ − M₂); negative t means first < second. For paired tests, direction depends on how the difference score is defined.

ANOVA

analysis of variance comparing means

Omnibus test

overall F test for any difference

Post hoc tests

follow-up comparisons identifying which means differ

Factor

IV

Level

category/condition of an IV.

Cell

one combination of factor levels

MS

mean square = SS/df

Main effect

effect of one IV averaged over the other IV

Interaction

one IV’s effect depends on the level of ano

Factorial design

study with two or more IVs

In what research situation is it most appropriate to perform a one-way ANOVA?

One IV with three or more independent groups/conditions being compared on one continuous DV (e.g., three exercise groups).

What risk does ANOVA protect from?

Inflated Type I error from running multiple t-tests on several group means.

How do you determine a true experiment from a quasi-experiment?

True experiment: researcher manipulates IV and randomly assigns participants to conditions. Quasi-experiment: lacks random assignment and/or uses preexisting groups.

State the null & alternative hypotheses for a single-factor (one-way) ANOVA.

H₀: μ₁ = μ₂ = μ₃ = … (all population means equal). H₁: at least one population mean differs (not necessarily every mean).

What does the omnibus test measure?

Tests whether there is any statistically significant difference among the group means; does not identify which groups differ.

What are the sources of variance for ANOVA? What are the sources of between-group variance? What are the sources of within-group variance?

Total variability = between-groups + within-groups. Between: treatment effects + random error/individual differences. Within: individual differences and random/experimental error (not treatment effects in a simple between-subjects design).

If the null hypothesis is true, what is the expected value of F?

F ≈ 1, because between-group and within-group mean squares should estimate the same error variance when H₀ is true.

What measure of effect size is used for ANOVA and how do you compute it?

Eta-squared (η²) = SSbetween / SStotal; proportion of total variance explained by the factor. Multiply by 100 for a percentage. Repeated-measures/factorial output may also report partial η².

What are the guidelines for determining the size of η2 (e.g., small, medium, large)?

η²: .01 = small; .06 = medium; .14 or greater = large.

What degrees of freedom are used to determine a critical value for an F ratio?

Numerator df = dfbetween = k − 1; denominator df = dfwithin = N − k (one-way independent ANOVA). Both and α determine critical F.

When is it appropriate to compute post hoc tests (Scheffe, etc.)?

After a significant omnibus F (p < .05), especially with 3+ levels, to discover which specific pairs of means differ.

How can you determine which group has a higher score when using post hoc tests?

Compare the group means and the sign/order of the reported mean difference; the group with the larger M scored higher. Check adjusted post hoc p for significance.

In an ANOVA, the F ratio is a ratio of what?

F = MSbetween / MSwithin (or MStreatment / MSerror): estimated treatment-plus-error variance over error variance.

What is the shape of the distribution of F-ratios?

An F distribution is nonnegative and right-skewed, with a long right tail; its exact shape depends on both dfs.

How do you interpret an R output for a one-way ANOVA? What values should be reported in APA format?

Read the factor row: df for factor and residuals, F, and p (Pr(>F)); p < .05 means at least one group differs. Report F(dfbetween, dfwithin), p, η², group M/SD, and significant post hoc comparisons. Example: F(2, 27) = 4.25, p = .025, η² = .24.

In what research situation is it appropriate to perform repeated measures ANOVA?

Same participants measured in 2+ conditions/times; especially used for three or more related conditions (two can be compared with a paired t-test).

What is the two-stage division of variance for a repeated measures ANOVA?

Stage 1: SStotal = SSbetween treatments + SSwithin treatments. Stage 2: SSwithin = SSindividual differences + SSerror. Remove individual differences from the denominator of F.

In what research situation is it appropriate to perform a factorial ANOVA?

When testing two or more independent variables simultaneously on the same DV, including their possible interaction

What is the factorial notation for a study with 2 factors and 3 levels in each factor? How many samples would you need?

3 × 3 factorial design = 2 factors, 3 levels each; 3 × 3 = 9 cells/condition samples (not necessarily just 9 participants)

What are the characteristics of a factorial design?

Two or more IVs; 2+ levels per IV; all combinations of levels form cells; estimates main effects and interactions.

What is the division of variance for a two-factor ANOVA?

SStotal = SSA + SSB + SSA×B + SSwithin/error. Between-cells variability is separated into factor A’s main effect, factor B’s main effect, and interaction.

What is the simplest possible factorial design? What is the factorial notation for this design?

2 × 2 design: two IVs with two levels each, producing four cells.

How many independent variables are needed for a two-factor design?

Exactly two IVs (factors).

If the results of a two-factor ANOVA are presented in a graph, what would indicate the presence of an interaction between variables?

Nonparallel lines (especially crossing or diverging/converging lines) suggest interaction. Parallel lines suggest no interaction.

State the null & alternative hypotheses for a main effect in a two-factor ANOVA.

For factor A: H₀: marginal means across A levels are equal (no A main effect); H₁: at least one differs. Same statements separately for factor B.

State the null & alternative hypotheses for an interaction in a two-factor ANOVA.

H₀: no A × B interaction; effect of A is the same at all B levels. H₁: A × B interaction exists; effect of A depends on B.

What are the three advantages of a factorial design?

Economy (more information for similar work); experimental control (include/control additional influences); ability to examine interactions.

How do we decide whether to reject or fail to reject the null hypothesis for a main effect in a two-way ANOVA?

Check the p-value for each main-effect F ratio: p < .05 → reject that factor’s H₀; p ≥ .05 → fail to reject. Each factor is tested separately.

How do we decide whether to reject or fail to reject the null hypothesis for an interaction in a two-way ANOVA?

Check the interaction A × B row: p < .05 → reject no-interaction H₀; p ≥ .05 → fail to reject. Interpret significant interactions before discussing main effects alone.

How many F ratios are obtained for a two-factor ANOVA?

Three: one F for factor A, one for factor B, and one for A × B interaction (all use MSerror as denominator in standard two-way ANOVA).

How do you interpret an R output for a factorial ANOVA? What values should be reported in APA format?

Find factor A, factor B, and A:B (interaction) rows in R’s ANOVA table. For each, read df, F value, and p (Pr(>F)); determine significance separately. Report means/SD, F(df_effect, df_error), p, effect sizes (η² or partial η²), and any post hoc/simple-effects tests as appropriate. Example: A: F(1, 36) = 5.42, p = .026; B: F(1, 36) = 0.72, p = .40; A × B: F(1, 36) = 4.10, p = .050. (Illustrative values only.)