Two samples from the same population both have M = 84 and s2 = 20, but one sample has n = 10 and the other has n = 20 scores. Both samples are used to evaluate a hypothesis stating that μ = 80 and to compute Cohen’s d. How will the outcomes for the two samples compare?

Question

​Two samples from the same population both have M = 84 and s2 = 20, but one sample has n = 10 and the other has n = 20 scores. Both samples are used to evaluate a hypothesis stating that μ = 80 and to compute Cohen’s d. How will the outcomes for the two samples compare?

BlackTom AI · Accepted Answer

When comparing two samples from the same population with the same reported statistics, it helps to separate the effect size from the test statistic.

First, consider Cohen’s d. It is computed as (Xbar − μ0) / s, where s is the standard deviation (the square root of the variance). Here Xbar = 84, μ0 = 80, and s^2 = 20, so s = sqrt(20) ≈ 4.472. Thus d ≈ (84 − 80) / 4.472 ≈ 4 / 4.472 ≈ 0.895. Since these values do not depend on sample size n (they rely on the mean difference and the standard deviation), both samples yield essentially the same Cohen’s d value even though their n differs.

Next, turn to hypothesis testing. The test statistic for testing μ = μ0 with known (or estimated) s is t = (Xbar − μ0) / (s / sqrt(n)). Here the numerator (Xbar − μ0) is the same across samples (84 − 80 = 4), and s is also the same (sqrt(20)). However, because the standard error is s / sqrt(n), the sample with larger n (n = 20) has a smaller standard error, leading to a larger absolute t value and thus a greater likelihood of rejecting H0 if the true mean is 80. Conversely, the smaller sample (n = 10) will have a larger standard error and a smaller t value, making rejection less likely.

Therefore, the statement that "the larger sample is more likely to reject the hypothesis" is true, while the claim that "the two samples will have the same value for Cohen’s d" is also true. The reasoning hinges on the distinction between a standardized effect size (Cohen’s d), which is insensitive to n in this setup, and the test statistic (t), which scales with sqrt(n) and thus benefits from a larger sample size.

Two samples from the same population both have M = 84 and s2 = 20, but one sample has n = 10 and the other has n = 20 scores. Both samples are used to evaluate a hypothesis stating that μ = 80 and to compute Cohen’s d. How will the outcomes for the two samples compare?Single choice

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​Two samples from the same population both have M = 84 and s2 = 20, but one sample has n = 10 and the other has n = 20 scores. Both samples are used to evaluate a hypothesis stating that μ = 80 and to compute Cohen’s d. How will the outcomes for the two samples compare?Single choice