What do these terms mean, and why are they important?
To understand the two terms, we must re-visit the normal curve:
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.
The same concept is depicted in the figure below (showing both examples of a one-tailed test):
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).