7 and 8

Independent Samples t-Test Null Hypothesis (H₀)

H₀: μ₁ - μ₂ = 0 (no difference between population means).

Independent Samples t-Test Alternative Hypothesis (H₁)

H₁: μ₁ - μ₂ ≠ 0 (two-tailed; population means are different).

Degrees of Freedom for Independent Samples t-Test

df_total = (n₁ - 1) + (n₂ - 1) = n₁ + n₂ - 2.

Pooled Variance (s²_p) Formula

s²_p = (SS₁ + SS₂) / (df₁ + df₂).

Estimated Standard Error of Difference between Two Independent Means

s(x̄₁ - x̄₂) = √[(s²_p / n₁) + (s²_p / n₂)].

95% Confidence Interval for Independent Samples t-Test

CI₉₅ = (x̄₁ - x̄₂) ± [t_crit(df) × s(x̄₁ - x̄₂)].

Statistical Significance and 95% Confidence Intervals

If the 95% CI contains 0, the difference is not statistically significant.

Boxplot Features

Line = median; Box = IQR (25th-75th percentiles); Whiskers = 1.5×IQR; Points = outliers.

Independent Samples t-Test Normality Assumption

Populations are normal. If biased/skewed, outcome variable transformations must apply to both groups.

Homogeneity of Variance Assessment

Check equal variances using ratio s²_max / s²_min; should be < 4 (equal n) or < 2 (unequal n).

Welch Correction

Unpooled t-test with adjusted (lowered) degrees of freedom used when sample variances are unequal.

Cohen's d Formula (Independent Samples t-Test)

Cohen's d = (x̄₁ - x̄₂) / √s²_p.

Cohen's d Interpretation

Measures difference in standard deviations: ~0.2 small, ~0.5 medium, ~0.8 large.

Alternative Names for Repeated Measures t-Test

Correlated measures t-test, paired-samples t-test, matched design, before-and-after / pre-post.

Source of Dependency in Repeated Measures

Scores either belong to the exact same participants or paired/matched individuals.

Difference Score (D_i) Formula

D_i = X_postscore - X_prescore (or X_prescore - X_postscore).

Repeated Measures t-Test Null Hypothesis (H₀)

H₀: μ_D̄ = 0 (the average difference score in the population equals zero).

Repeated Measures t-Test Alternative Hypothesis (H₁)

H₁: μ_D̄ ≠ 0 (the population mean difference score differs from zero).

Degrees of Freedom for Repeated Measures t-Test

df = n - 1, where n is the number of difference scores (participants/pairs).

Standard Deviation of Difference Scores (s_D)

s_D = √[ Σ(D_i - D̄)² / (n - 1) ].

Standard Error of Mean Difference Scores (s_D̄)

s_D̄ = s_D / √n.

Repeated Measures t-Statistic Formula

t = (D̄ - μ_D) / s_D̄ = D̄ / s_D̄ (since μ_D = 0 under H₀).

95% Confidence Interval for Repeated Measures t-Test

CI₉₅ = D̄ ± [s_D̄ × t_crit(df)].

Cohen's d Formula (Repeated Measures t-Test)

Cohen's d = D̄ / s_D.

When to Calculate Cohen's d

Only after rejecting the null hypothesis.

Assumptions of Repeated Measures t-Test

Matched pairs data, independent sampling units, normally distributed mean difference scores.

Advantages of Repeated Measures t-Test

Increases power, partials out individual differences, reduces error variance.

Disadvantages of Repeated Measures t-Test

Carryover/order effects, fatigue, practice/memory effects, attrition.

Choosing Between One-Sample and Independent t-Tests

Use one-sample t-test for 1 group (μ known, σ unknown); independent t-test for 2 independent groups.

Choosing Between Independent and Dependent t-Tests

Use independent t-test if samples are unrelated; dependent t-test if samples are paired/repeated.

Primary Limitation of Multiple Pairwise t-Tests

Inflates Type I error rate (family-wise / experiment-wise error rate).

Family-Wise Error Rate Formula for k Independent Tests

α_familywise = 1 - (1 - α)^k.

Number of Pairwise Comparisons for k Groups

Total possible comparisons = k(k - 1) / 2.

Purpose of One-Way Analysis of Variance (ANOVA)

Compares means of 3+ groups simultaneously in one test without inflating α.

Two Variance Components in One-Way ANOVA

Mean Square Between (MSB) and Mean Square Within / Error (MSW).

Mean Square Between (MSB) Definition

Variance estimate based on differences between group means and the grand mean.

Degrees of Freedom Between Groups (df_between)

df_between = k - 1, where k is the number of groups.

Mean Square Within / Error (MSW) Definition

Variance estimate based on spread/variability of individual scores within each group.

Degrees of Freedom Within Groups (df_within)

df_within = N - k, where N is total sample size and k is group count.

F-Ratio Formula in ANOVA

F = MS_between / MS_within.

Expected F-Ratio Value Under Null Hypothesis

Approximately 1.0 (since MSB and MSW both estimate standard population variance).

Role of Post-Hoc Tests

Conducted after a significant ANOVA to identify specifically which group means significantly differ.

Directional Hypothesis Strategy in Dependent t-Tests

Setting H₁: μ_D > 0 places the entire α critical region in one tail, increasing statistical power.

Order of Operations for Dependent-Samples t-Test

1) Calculate difference scores (D_i), 2) Find D̄ and s_D, 3) Compute s_D̄, 4) Compute t_obt.

Decision Rule Strategy for One-Tailed t-Tests

Reject H₀ if t_obt ≥ t_crit in predicted direction; fail to reject if lower or opposite direction.

Theoretical Meaning of Sample Difference Variance (s²_D)

Measures individual variability in change scores across participants, directly impacting standard error.

Effect Size Strategy: Cohen's d vs t-Statistic

Cohen's d measures standardized mean difference independent of sample size, unlike t_obt.

Two-Way ANOVA Definition

Evaluates the effect of two independent variables (factors) on a single continuous dependent variable.

Main Effect in ANOVA

The mean effect of one independent variable on the dependent variable across all levels of other factors.

Interaction Effect in ANOVA

Occurs when the effect of one independent variable depends on the level of another independent variable.

Z-test vs. One-sample t-test

Z-test uses population standard deviation; t-test estimates it using sample standard deviation.

Degrees of freedom (df) for a one-sample t-test

df = n − 1

Estimated standard error of the mean formula

s / √n

Relationship between t-distribution and sample size

As sample size increases, the t-distribution approaches the standard normal Z-distribution.

One-sample t-test formula

t = (x̄ − μ) / sx̄

Cohen's d for a one-sample t-test

d = (x̄ − μ) / s

Cohen's d small effect size threshold

0.2

Cohen's d medium effect size threshold

0.5

Cohen's d large effect size threshold

0.8

Null hypothesis for a two-independent-samples t-test

H0: μ1 − μ2 = 0

Pooled variance (s²p) formula

s²p = (SS1 + SS2) / (df1 + df2)

Degrees of freedom (df) for two independent samples

df = n1 + n2 − 2

Standard error of the difference between two means

√((s²p / n1) + (s²p / n2))

Homogeneity of variance assumption

The assumption that both compared populations have equal variances.

Welch Correction

An unpooled t-test adjustment used when population variances are unequal.

Independent samples t-test formula

t = (x̄1 − x̄2) / s(x̄1 − x̄2)

What Cohen's d represents

The standardized difference between two means, measured in standard deviation units.

Unit of analysis for an independent-samples t-test

The difference between two sample means (X₁ - X₂).

Why is the t-distribution a "family" of distributions?

There is a different t-distribution shape for every degree of freedom (N - 1).

Pooled variance (s²p) conceptual definition

A weighted average of the individual group variances.

Confidence interval formula for a one-sample t-test

CI = x̄ ± (tcrit × sx̄)

Confidence interval formula for an independent-samples t-test

CI = (x̄₁ - x̄₂) ± (tcrit × s(x̄₁ - x̄₂))

When is a confidence interval typically omitted in NHST?

For one-tailed hypothesis tests.

Four steps of NHST for independent-samples t-test

1) Specify hypotheses, 2) Set decision criteria/null distribution, 3) Calculate test statistic, 4) Make binary decision.

SFU sample statistics in the satisfaction scenario

n = 10, mean = 7.2, sum of squared deviations = 67.6, variance = 7.51, SD = 2.74.

UBC sample statistics in the satisfaction scenario

n = 10, mean = 4.6, sum of squared deviations = 64.4, variance = 7.15, SD = 2.67.

Degrees of freedom calculation in the SFU vs. UBC scenario

df_total = 10 + 10 − 2 = 18.

Critical t-value (t_crit) for α = .05 (two-tailed, df = 18)

t_crit = ±2.101.

Pooled variance (s²_p) result in SFU vs. UBC scenario

s²_p = [9(7.51) + 9(7.15)] / 18 = 7.33.

Standard error (s(x̄₁−x̄₂)) result in SFU vs. UBC scenario

s(x̄₁−x̄₂) = √(7.33 × (1/10 + 1/10)) = 1.21.

Obtained t-statistic (t_obt) in SFU vs. UBC scenario

t_obt = (7.2 − 4.6) / 1.21 = 2.15.

Binary hypothesis decision in SFU vs. UBC scenario

Reject H₀ because t_obt (2.15) is more extreme than t_crit (2.101).

Calculated effect size (Cohen's d) for SFU vs. UBC scenario

d = 2.6 / √7.33 = 0.96 (a large effect).

95% Confidence Interval for mean difference (SFU vs. UBC)

CI_95% = 2.6 ± (1.21 × 2.101) = [.06, 5.14].

Why t-distributions have heavier tails than the Z-distribution

Using sample standard deviation adds extra uncertainty, increasing tail probability for extreme values.

Role of the Central Limit Theorem in t-tests

It ensures sampling distributions of means approximate normality as sample size grows.

Theoretical purpose of pooling variance

It combines group information to get a better overall population variance estimate.

Consequence of violating homogeneity of variance

It distorts Type I error rates, making standard t-tests unreliable.

Why effect size is necessary alongside hypothesis testing

Hypothesis tests show if an effect exists; effect size measures practical magnitude.

How larger sample size affects t-test power

Larger samples reduce standard error, making true population differences easier to detect.

Theoretical meaning of degrees of freedom in variance estimation

The number of values free to vary after sample mean calculation.

Shared assumptions of Z-tests and T-tests

Random sampling, independent observations, and normal population distribution (or large sample).

Z-test vs T-test distribution type

Z-test uses the standard normal distribution; T-test uses the t-distribution.

Shape flexibility of t-distribution vs Z-distribution

Z-distribution has a fixed shape; t-distribution shape changes based on degrees of freedom.

Main reason t-tests are used over z-tests in research

Population standard deviation (σ) is almost always unknown in real research.

Standard deviation parameter difference in formulas

Z-test formula uses population SD (σ); t-test formula uses sample SD (s).

When to use a one-sample t-test

Comparing sample mean to population mean when population standard deviation (σ) is unknown.

Step 1: Given information needed for a one-sample t-test

Population mean (μ), sample mean (X̄), sample SD (s), sample size (n), alpha (α).

Step 2: Key phrases for a two-tailed test hypothesis

"Different," "affect," or "change" (H₀: μ = μ₀, H_A: μ ≠ μ₀).

Step 2: Key phrase for a right-tailed test hypothesis

"Greater than" (H_A: μ > μ₀).