One-Tailed And Two-Tailed Test

When the null hypothesis is true, the t statistic follows the t-distribution.

The t-distribution is similar to a Normal distribution. The probability of getting a value close to zero is very high, while the probability of getting a value far from zero is low. So if the null hypothesis is true, it is very unlikely to get the value of t far from zero. If this happens, the null hypothesis is rejected and the alternative one is accepted.

Critical region

Highlighted in red is the critical region (or rejection region). When the t-statistic falls within this critical region, the null hypothesis is rejected, and the alternative hypothesis is accepted.

The critical region is chosen so that the probability of the t-statistic landing inside it equals the significance level, typically set at α (usually 0.05).

One-Tailed vs Two-Tailed

Depending on the alternative hypothesis, there are two methods to construct a critical region.

• A two-tailed test is used when the alternative hypothesis is "Means are not equal.";
• A one-tailed test is used when the alternative hypothesis is "One mean is greater (lower) than the other."

Example

If the t statistic for the comparison of male and female heights is computed and found to be 19.1, it falls within the critical region. This allows the conclusion that males are statistically significantly taller than females.

In this example, any value greater than 1.65 falls within the critical region. This is known as a critical value. The critical value is influenced by the sample sizes, but there's no need to concern yourself with it. Python will calculate both the critical value and the t statistic for you.