Final answer:
The standard error of the difference in two proportions (SE(D1-D2)) is calculated using sample proportions and sizes, whereas for sample means, the formula involves the means and sample standard deviations. Cohen's d is a measure of effect size involving the difference between means and pooled standard deviation. A t-score is derived by standardizing the difference between sample means using the standard error.
Step-by-step explanation:
To find the standard error of the difference in two proportions, SE(D1-D2), assuming you're dealing with sample proportions rather than means, you can use a modified formula:
SE(D1-D2) = √(p1(1-p1)/n1 + p2(1-p2)/n2)
where p1 and p2 are the sample proportions, and n1 and n2 are the sizes of the two independent samples. However, it looks like there may be some confusion, as the information you've provided talks about the standard error of the difference in sample means, which is similar, but involves means (X₁ and X₂) and sample standard deviations (S1 and S2).
For the means, the standard error can be found using the formula:
SE(X₁ - X₂) = √((S1^2)/n1 + (S2^2)/n2)
This accounts for the variability in both samples and allows you to standardize the difference between the two means to obtain a t-score test statistic if the population standard deviations are unknown and the sample sizes are sufficiently large.
Cohen's d
Cohen's d is calculated by taking the difference between two means and dividing by the pooled standard deviation:
Cohen's d = (X1 - X2) / Sₚooled
The pooled standard deviation is a weighted average of the two sample standard deviations, which can be used as an estimate of the common standard deviation of the two samples if the population standard deviations are assumed equal.