One Sample T- Test

The one-sample t-test checks whether a sample mean is genuinely different from a known or claimed population value. You draw a single random sample, compute its mean, and compare that mean against the value you're testing against — deciding, on the strength of the sample alone, whether the population really differs from that benchmark.

The test statistic is:

where $\mu_0$ is the value being tested against (the "comparison value"), $\bar{x}$ is the sample mean, $S$ is the sample standard deviation, and $n$ is the sample size. The denominator, $S/\sqrt{n}$, is the standard error of the mean — it scales the gap between $\bar{x}$ and $\mu_0$ by how much sample means naturally vary from sample to sample. …