For Case 2A, which expression defines Tcalc?

For Case 2A, which expression defines Tcalc?

• A. Tcalc = (|mean1 - mean2| / spooled) * sqrt((n1*n2)/(n1+n2))
• B. Tcalc = (mean1 - mean2) / sqrt((s1^2/n1)+(s2^2/n2))
• C. Tcalc = (mean1 - mean2) / spooled
• D. Tcalc = (|mean1 - mean2|) / sqrt(n1+n2)

Answer: (A)

In Case 2A, you compare two independent samples assuming equal population variances, so you use a pooled standard deviation to standardize the difference in means. The standard error of the difference is spooled times sqrt(1/n1 + 1/n2), where spooled is the pooled standard deviation computed from both sample variances. Therefore the t statistic is Tcalc = (mean1 - mean2) / (spooled * sqrt(1/n1 + 1/n2)). This is mathematically equivalent to Tcalc = (|mean1 - mean2| / spooled) * sqrt(n1 n2 /(n1 + n2)). The expression that uses the absolute mean difference divided by the pooled SD and multiplied by sqrt(n1 n2 /(n1 + n2)) matches this form. Why other forms don’t fit: using s1^2/n1 + s2^2/n2 inside the square root corresponds to a version that does not assume equal variances (Welch’s approach). Omitting the sqrt term with 1/n1 + 1/n2 or using sqrt(n1 + n2) in the denominator disrupts the correct standard error for the difference in means under the equal-variances assumption.

In Case 2A, you compare two independent samples assuming equal population variances, so you use a pooled standard deviation to standardize the difference in means. The standard error of the difference is spooled times sqrt(1/n1 + 1/n2), where spooled is the pooled standard deviation computed from both sample variances. Therefore the t statistic is Tcalc = (mean1 - mean2) / (spooled * sqrt(1/n1 + 1/n2)).

This is mathematically equivalent to Tcalc = (|mean1 - mean2| / spooled) * sqrt(n1 n2 /(n1 + n2)). The expression that uses the absolute mean difference divided by the pooled SD and multiplied by sqrt(n1 n2 /(n1 + n2)) matches this form.

Why other forms don’t fit: using s1^2/n1 + s2^2/n2 inside the square root corresponds to a version that does not assume equal variances (Welch’s approach). Omitting the sqrt term with 1/n1 + 1/n2 or using sqrt(n1 + n2) in the denominator disrupts the correct standard error for the difference in means under the equal-variances assumption.