Which of the following is the formula for the test statistic used in a test of paired samples?

• A. $\frac{x-y}{\sqrt{\frac{s{1}^2}{n}}}$
• B. $\frac{∆-0}{\frac{s}{\sqrt{n}}}$
• C. $\frac{p-q}{\sqrt{\frac{p\times q}{n}}}$
• D. $\frac{x-\mu }{\frac{\sigma }{\sqrt{n}}}$

Answer: (B)

Verified step by step guidance

1. Identify that the problem is asking for the formula of the test statistic used in a paired samples test, which compares the means of two related groups.
2. Recall that in a paired samples test, we analyze the differences between paired observations, often denoted as \( \Delta = x - y \), where \( x \) and \( y \) are paired sample values.
3. The test statistic formula for paired samples is based on the mean difference \( \bar{d} \) minus the hypothesized difference (usually 0), divided by the standard error of the mean difference.
4. Express the test statistic as: \[\frac{\bar{d} - 0}{\frac{s_d}{\sqrt{n}}}\] where \( \bar{d} \) is the sample mean of the differences, \( s_d \) is the sample standard deviation of the differences, and \( n \) is the number of pairs.
5. Understand that this formula measures how many standard errors the observed mean difference is away from the hypothesized mean difference, which is the basis for conducting hypothesis tests on paired data.