If a study with a large sample (e.g., 500 participants per group) yields a strong relationship (such as Cohen's $d = 0.50$), what is the typical statistical conclusion regarding the null hypothesis?

Researcher X finds a strong relationship ($d = 0.45$) in a sample of $400$ men and $400$ women. Researcher Y finds a weak relationship ($d = 0.05$) in a sample of $4$ men and $4$ women. Because Researcher Y's result is more likely to have occurred by chance, Researcher Y should reject the null hypothesis.

Evaluate the following research scenarios and arrange the conclusions in order from the least scientifically sound (most likely to be a result of chance) to the most scientifically sound (least likely to be a result of chance).

Suppose you are tasked with synthesizing two prototypical research examples for an introductory psychology manual to illustrate how sample size and relationship strength dictate statistical conclusions. Which pair of study configurations and outcomes should you construct to best demonstrate when a researcher is justified in rejecting versus retaining the null hypothesis?

A strong relationship found in a large sample ($N = 1,000$, $d = 0.50$) results in the rejection of the null hypothesis because it is highly improbable to occur by chance, whereas a weak relationship in a small sample ($N = 6$, $d = 0.10$) is retained because it is statistically probable to occur even if no real effect exists.

In the context of hypothesis testing, what is the most likely statistical conclusion if a study comparing two groups uses a very small sample (e.g., 3 individuals per group) and yields a weak relationship (e.g., Cohen's d = 0.10)?

Imagine you are reviewing three different psychology studies that compare two groups. Based on the relationship between sample size and relationship strength (Cohen's d), rank these studies from the one most likely to result in rejecting the null hypothesis (top) to the one least likely to result in rejecting the null hypothesis (bottom).

A researcher claims to have discovered a gender difference based on a study of $3$ women and $3$ men that yielded a weak relationship ($d = 0.10$). From a statistical perspective, this conclusion would be evaluated as unreliable because such a result is quite likely to occur by _____.

A researcher rejects the null hypothesis for a study with $500$ participants and a Cohen's $d = 0.50$ because the result is highly unlikely if the null is true; conversely, a researcher retains the null hypothesis for a study with $3$ participants and a $d = 0.10$ because that result is quite _____ to occur by chance.

Match each research scenario to the most likely null hypothesis testing outcome based on the study's sample size and effect size.

Match each description on the left with the corresponding statistical outcome or reasoning on the right.

Explain how sample size and the strength of a relationship (such as Cohen's $d$) jointly influence the decision to reject or retain the null hypothesis in a study comparing two groups. Use the two hypothetical studies from the text—one with a large sample of $500$ women and $500$ men showing $d = 0.50$, and another with a small sample of $3$ women and $3$ men showing $d = 0.10$—to illustrate your explanation.

Based on the principles of sample size and relationship strength, diagnose what statistical decision the psychologist should make regarding the null hypothesis for both Pilot Study A and Pilot Study B, and justify each decision based on the likelihood of the findings occurring by chance.

Analyze why obtaining a Cohen's $d = 0.10$ in a sample of $3$ women and $3$ men leads to a different statistical conclusion than obtaining a Cohen's $d = 0.50$ in a sample of $500$ women and $500$ men.