Can you provide an example of standard error's application in hypothesis testing?

In hypothesis testing, the standard error is used to calculate test statistics (like t-scores or z-scores) that compare a sample statistic to a hypothesized population parameter. For example, when testing a hypothesis about a population mean, the t-statistic is calculated as (Sample Mean - Hypothesized Mean) / Standard Error of the Mean. If the calculated t-statistic is significantly large (i.e., falls into the rejection region determined by the significance level), it suggests that the observed difference is unlikely to be due to random chance, allowing us to reject the null hypothesis. The standard error acts as a scaling factor, allowing us to judge the difference relative to the variability of the sample estimate.