The t-value is a test statistic for t-tests that measures the difference between an observed sample statistic and its hypothesized population parameter in units of standard error. A t-test compares the observed t-value to a critical value on the t-distribution with (n-1) degrees of freedom to determine whether the difference between the estimated and hypothesized values of the population parameter is statistically significant.

Applications of t-values include:

- comparing two sample means
- comparing the means of paired observations
- determining the significance of a regression coefficient
- comparing two regression coefficients

You can also use t-values in a 1-sample t-test. For example, you want to determine whether the length of a manufactured part meets its target value of 10cm. You take a sample of 50 parts, conduct a two-sided 1-sample t-test on their mean length with the following hypotheses:

- H
_{0}: μ = 0 (the mean length of all parts meets the target value ) - H
_{1}: μ ≠ 0 (the mean length of all parts does not meet the target value)

The test produces a t-value of 2.5. On the t-distribution with (n-1 = 49) degrees of freedom, this t-value corresponds to a p-value of 0.0158. For most common significance levels, this result is statistically significant. Therefore, you reject the null hypothesis that the mean length meets the target, and conclude that the process needs improvement.