Question

The test statistic $t$ for testing the significance of differences between the means of two independent samples is given by

• A. $t = \dfrac{\bar{x} - \bar{y}}{\sqrt{s}}$
• B. $t = \dfrac{\bar{x} - \bar{y}}{s \sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
• C. $t = \dfrac{\bar{x} - \bar{y}}{s / \sqrt{n - 1}}$
• D. $t = \dfrac{\bar{x} + \bar{y}}{s \sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$

Hint:

Use the two-sample $t$-test formula when comparing means of independent samples with assumed equal variance.

Answer 1

The Correct Option is B

To assess the disparity between the averages of two distinct, independent groups, a two-sample $t$-test is employed.
The formula for the test statistic is:
$t = \dfrac{\bar{x} - \bar{y}}{s \sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
In this equation, $\bar{x}$ and $\bar{y}$ represent the means of the respective samples, $s$ denotes the pooled standard deviation, and $n_1$ and $n_2$ are the sizes of the samples.
This calculation incorporates the variability from both samples and is applicable under the assumption of equal population variances.
Specifically, the numerator quantifies the difference between the sample means, while the denominator modifies this difference by the standard error.
Therefore, option (B) is the correct choice.