Tag Archives: Usain Bolt

One-tailed and two-tailed tests: Part 1

Often, one encounters the term “one-tailed test” and “two-tailed test” in scientific literature.

What do these terms mean, and why are they important?

To understand the two terms, we must re-visit the normal curve:

The Standard Normal Curve showing the area under the curve at 1 Standard Deviation intervals from the mean

The various numbers under the curve indicate the area under the curve as a fraction of the total. Since the total area under the curve is one (1), the area under each part of the curve is a fraction of one.

In hypothesis testing, we typically choose an alpha error of 5%. That is, the probability of rejecting a null hypothesis when it is actually true is 5%. This is also called the false positive error rate.

Effectively, we wish to capture 95% of the area under the normal curve.

This could be done in one of three ways:

1. The entire 5% could be on the right side of the mean;

2. The entire 5% could be on the left side of the mean;

3. The 5% could be split into two equal halves, with 2.5% on either side of the mean.

Usually, we choose options 1 or 2 when we are fairly certain that the results of the study would go a particular way.

Example: In a 100 m race between myself and Usain Bolt, it is obvious that Usain would win the race.

In hypothesis testing terms, we would state the null and alternate hypotheses as follows:

Null Hypothesis (H0): Usain Bolt will not win a 100 m race against Dr. Roopesh

Alternate Hypothesis (Ha): Usain Bolt will win the 100 m race against Dr. Roopesh

Here we are clearly predicting the direction of the outcome.

The above is an example of a one-tailed hypothesis test.

Figure showing a right-tailed test (values falling in the deep blue shaded area will lead to a rejection of the null hypothesis)

The same concept is depicted in the figure below (showing both examples of a one-tailed test):

Figure showing both a right and left tailed test with acceptance and rejection regions, and the Alternate hypotheses (H1)

As can be seen, the shaded area corresponds to 5% of the area under the curve; but in either case, the shaded area is to one side of the midline (mean).