One-tailed and two-tailed tests

The example in the previous section was an instance of a one-tailed test where the null hypothesis is rejected or accepted based on one direction of the normal distribution.

In a two-tailed test, both the tails of the null hypothesis are used to test the hypothesis.

In a two-tailed test, when a significance level of 5% is used, then it is distributed equally in the both directions, that is, 2.5% of it in one direction and 2.5% in the other direction.

Let's understand this with an example. The mean score of the mathematics exam at a national level is 60 marks and the standard deviation is 3 marks.

The mean marks of a class are 53. The null hypothesis is that the mean marks of the class are similar to the national average. Let's test this hypothesis by first getting the z-score 60:

>>> zscore = ( 53 - 60 ) / 3.0
>>> zscore
-2.3333333333333335

The p-value would be:

>>> prob = stats.norm.cdf(zscore)
>>> prob

0.0098153286286453336

So, the p-value is 0.98%. The null hypothesis is to be rejected, and the p-value should be less than 2.5% in either direction of the bell curve. Since the p-value is less than 2.5%, we can reject the null hypothesis and clearly state that the average marks of the class are significantly different from the national average.