Two-Sample t-Tests

What are the two main factors that affect test selection in statistics?

Research design and whether population parameters (μ, σ²) are known or assumed.

When do we use a z-test?

When comparing a single group to a population and both μ and σ² are known or assumed.

Why is the t-distribution used instead of the z-distribution in most practical research?

Because population variance is usually unknown and must be estimated from sample data.

What is the formula for estimated population variance?

σ^2=s2=∑(Xi−xˉ)2

—————————

N−1

What are degrees of freedom in variance estimation?

The number of independent data points after estimating parameters, typically N - 1.

What is a t-distribution?

A family of distributions used when estimating population variance, with shape dependent on degrees of freedom.

What are key differences between t-distributions and the z-distribution?

t-distributions have heavier tails and use varying critical values based on degrees of freedom.

What is the standard error formula in a one-sample t-test?

SE= s

————————

(square rooted) N​​

What is the test statistic formula for a one-sample t-test?

t= xˉ−μ

————

s/ (sqaure rooted) N​

How is Cohen’s d calculated in a one-sample t-test?

d=xˉ−μ

———-

s​

What is a dependent-samples t-test used for?

Comparing the same group at two time points (within-subjects design).

How are difference scores calculated in a dependent-samples t-test?

D=Time 2−Time 1

What is the test statistic for dependent-samples t-test?

t=Dˉ−μD

————

sD/(square rooted) N​​

What is typically the null hypothesis for dependent-samples t-tests?

μD​=0, meaning no change over time.

What is the 7-step NHST procedure for dependent-samples t-test?

Define population, state hypotheses, check assumptions, define test statistic, set α and critical values, calculate stats, make decision.

What does an independent-samples t-test compare?

Two independent groups on the same outcome measure (between-subjects design).

What is the null hypothesis in an independent-samples t-test?

μ1​−μ2​=0, meaning no difference between the two groups.

What is the pooled variance formula?

How is standard error for independent samples calculated?

SE=(square root all) sp2 sp2

——​​ + ——

N1 N2​

What is the test statistic for independent-samples t-test?

t=(xˉ1−xˉ2)−(μ1−μ2)

—————————-

SE​

How is Cohen’s d calculated for independent samples?

d=​xˉ1​−xˉ2​​

—————-

sp

What are assumptions for the independent-samples t-test?

Normality in both populations and homogeneity of variances.

What is the confidence interval formula for independent samples?

CI=(xˉ1−xˉ2)±(tcrit×SE)