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, we can reject the null hypothesis and accept the alternative one.

Critical region

Highlighted in red is the critical region (or rejection region). When the t-statistic's value falls within this critical region, we reject the null hypothesis and accept the alternative hypothesis.

We choose the critical region in such a way that the probability of landing inside it is equivalent to 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 we compute the t statistic for our example comparing male and female heights, we obtain a value of 19.1. Since it falls within a critical region, we can conclude 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.