PSYC 210 Lecture 7: One-Sample and Two-Sample t-Tests

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).

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

"Less than" (H_A: μ < μ₀).

Step 3: Degrees of freedom formula for one-sample t-test

df = n − 1

Step 4: Standard error (SE) formula for one-sample t-test

SE = s / √n

Step 5: Formula for calculated/obtained t-value

t = (X̄ − μ) / SE

Step 6: Inputs required to find critical t-value in t-table

Alpha level (α), degrees of freedom (df), and number of tails (one vs. two).

Step 7: How to compare obtained t-value to critical t-value

Ignore the sign and compare absolute values (|t_obt| vs. |t_crit|).

Step 8: Decision rule to reject H₀

Reject H₀ if |t_obt| > |t_crit|.

Step 8: Decision rule to fail to reject H₀

Fail to reject H₀ if |t_obt| < |t_crit|.

Step 9: Wording when rejecting H₀

There is sufficient evidence to conclude a significant effect exists.

Step 9: Wording when failing to reject H₀

There is insufficient evidence to conclude a significant effect exists.

Memory Trick acronym for one-sample t-test steps

S-H-D-S-T-C-D-C (Statistics, Hypotheses, df, SE, t-obt, t-crit, Decision, Conclusion).

Relationship between independent-samples and two-sample t-tests

An independent-samples t-test is a specific type of two-sample t-test used for unrelated groups.

Types of two-sample t-tests

Two-sample t-tests include independent-samples t-tests (unrelated groups) and paired-samples t-tests (matched/repeated measures).

Independent-samples vs. Paired-samples t-test target data

Independent-samples tests compare separate groups; paired-samples tests compare two measurements from the same or matched subjects.

Core assumption of independent-samples t-tests

Observations between and within the two groups must be completely independent.

Sample size requirement difference

Independent-samples t-tests can have different group sizes (n₁ ≠ n₂); paired-samples t-tests require equal sample sizes.