PSYC 301: Module 2

What is the core function of statistics according to Module 2?

Statistics are used to aid in formulating good arguments that explain observed comparative differences, not merely to calculate numbers

What is the simplest explanation for an observed difference?

Chance — it is the baseline explanation assumed unless the data require a more complex explanation involving systematic factors

What explanation do we adopt only if chance is insufficient?

A combined explanation involving both chance and systematic influences

Why do sample means differ even if populations do not?

Because of sampling error — random samples rarely perfectly represent the population, causing slight differences in sample means

What is sampling error?

Random variation in sample composition that causes sample statistics to differ from population parameters, even when no true population difference exists

How common are large sample differences due to chance alone?

They are rare; most samples produce small differences, while large discrepancies occur infrequently

What role does NHST play in evaluating chance?

NHST tells us how rare an observed difference would be if chance were the only explanation

When is the chance explanation considered viable?

When the observed difference is common under the assumption that the population difference is zero

When do we doubt the chance explanation?

When the observed difference is very rare under the assumption of no population difference

Why is sample size important in NHST?

Sample size affects how well a sample represents the population and how much sampling error exists

How do small samples affect NHST conclusions?

Small samples have high sampling error, so only large differences are judged unlikely under chance

How do large samples affect NHST conclusions?

Large samples have low sampling error, so even modest differences may be judged unlikely under chance

In the study skills example, how many populations are involved?

Two populations:

1. Those who received study skills training

2. Those who did not

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

The population means are exactly equal (μ₁ = μ₂), meaning no true difference exists

What does “independent samples” mean?

The two samples are drawn from separate populations, and membership in one group precludes membership in the other

What statistical test is used to compare two independent group means?

Independent samples t-test (also called independent measures t-test)

What is the general conceptual formula for a t-test?

t = (sample data − hypothesized population parameter) / estimated standard error

What does a larger t value indicate?

A greater discrepancy between the observed data and what would be expected under the null hypothesis

What does the independent samples t-test evaluate?

Whether the difference between two sample means is likely due to chance or reflects a population mean difference

Why is the null difference (μ₁ − μ₂ = 0) dropped from the t formula?

Because the null hypothesis states there is no population difference, simplifying the formula to observed mean difference divided by standard error

What is standard error conceptually?

A measure of how accurate the sample mean(s) are as estimates of population means

How do dispersion and sample size affect standard error?

• Greater dispersion → larger error

• Larger sample size → smaller error

Why are two sources of error pooled in the independent samples t-test?

Because each group’s mean has its own sampling error that contributes to uncertainty in the mean difference

What are degrees of freedom for the independent samples t-test?

df = (n₁ − 1) + (n₂ − 1)

What factors influence the size of a t value?

• Magnitude of the mean difference

• Sample size

• Variability (dispersion) in the samples

Why can t values be interpreted probabilistically?

Because the t statistic has a known sampling distribution under the null hypothesis

How does the t distribution change with degrees of freedom?

• Few df → wider, flatter distribution

• Many df → narrower, approximates the normal distribution

What is alpha (α)?

The probability threshold for deciding when a chance explanation is no longer tenable (commonly .05)

Why is the default t-test two-tailed?

Because the test does not privilege a direction of effect unless specified a priori

What is a Type I error?

Rejecting the null hypothesis when it is actually true (false positive)

How is alpha related to Type I error?

Alpha represents the probability of committing a Type I error

What is a one-tailed test?

A test used when there is a strong directional prediction, placing all alpha in one tail of the distribution

Why are one-tailed tests considered more “liberal”?

Because less extreme values are required for statistical significance

What is a Type II error (β)?

Failing to reject the null hypothesis when it is false (false negative)

What is statistical power?

The probability of correctly rejecting a false null hypothesis (Power = 1 − β)

What factors determine power?

• Alpha level

• Sample size

• Effect size

Why is low power a serious problem?

It increases false negatives and false positives and contributes to the replication crisis

What assumptions underlie the independent samples t-test?

• Independence of observations

• Normality of outcome distributions

• Homogeneity of variance

What is the purpose of a repeated measures t-test?

To test mean differences using the same sample of participants

What data are analyzed in a repeated measures t-test?

Difference scores (D) between paired observations

What is the null hypothesis for a repeated measures t-test?

The mean difference score in the population is zero (μᴰ = 0)

What are the degrees of freedom for a repeated measures t-test?

df = n − 1

What are advantages of repeated measures designs?

• Greater power

• More economical use of participants

What are advantages of independent samples designs?

• No carryover effects

• Less vulnerability to demand characteristics