The t statistic is calculated using the formula that involves the sample mean, the population mean, the sample standard deviation, and the square root of the sample size. In this context, the correct formula, which is often used in hypothesis testing, is expressed as:
t = (x̅ - mean) / (s / SQRT(n)).
This formula indicates that the t statistic measures how far the sample mean (x̅) is from the population mean, taking into account the variability in the sample (expressed by the sample standard deviation, s) and the number of observations in the sample (n). The division by the square root of n (SQRT(n)) serves to adjust the standard deviation based on sample size, which is essential because larger samples tend to provide more reliable estimates of the population parameters.
Thus, the reason this formula is correct lies in its comprehensive approach to determining the t statistic, ensuring that the impact of sample size on variability is properly accounted for, which is crucial for accurate statistical inference.