The t-distribution is a probability distribution used when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, which accounts for the extra uncertainty.

The t-statistic measures how many standard errors the difference between two sample means is away from zero (the null hypothesis value). It is calculated based on the difference between sample means, the pooled standard deviation, and the sample sizes.

The critical region represents the range of t-values for which the null hypothesis is rejected at a chosen significance level (a). If the observed t-statistic falls inside this region (beyond the critical values), it suggests that the difference between means is statistically significant.

This visualization shows the two normal distributions, the t-distribution, the critical regions shaded in red, and the observed t-statistic as a dashed line. Adjust the parameters to see how these elements change.