5a) One and Two Tailed Tests
Suppose we have a null hypothesis H0 and an alternative hypothesis H1. We consider the distribution given by the null hypothesis and perform a test to determine whether or not the null hypothesis should be rejected in favour of the alternative hypothesis.
There are two different types of tests that can be performed. A one-tailed test looks for an increase or decrease in the parameter whereas a two-tailed test looks for any change in the parameter (which can be any change- increase or decrease).
We can perform the test at any level (usually 1%, 5% or 10%). For example, performing the test at a 5% level means that there is a 5% chance of wrongly ...view middle of the document...
We therefore reject the null hypothesis in favour of the alternative at the 5% level.
However, the probability is greater than 0.01, so we would not reject the null hypothesis in favour of the alternative at the 1% level.
In a two-tailed test, we are looking for either an increase or a decrease. So, for example, H0 might be that the mean is equal to 9 (as before). This time, however, H1 would be that the mean is not equal to 9. In this case, therefore, the critical region has two parts:
Lets test the parameter p of a Binomial distribution at the 10% level.
Suppose a coin is tossed 10 times and we get 7 heads. We want to test whether or not the coin is fair. If the coin is fair, p = 0.5 . Put this as the null hypothesis:
H0: p = 0.5
H1: p ≠ 0.5
Now, because the test is 2-tailed, the critical region has two parts. Half of the critical region is to the right and half is to the left. So the critical region contains both the top 5% of the distribution and the bottom 5% of the distribution (since we are testing at the 10% level).
If H0 is true, X ~ Bin(10, 0.5).
If the null hypothesis is true, what is the probability that X is 7 or above?
P(X ≥ 7) = 1 - P(X < 7) = 1 - P(X ≤ 6) = 1 - 0.8281 = 0.1719
Is this in the critical region? No- because the probability that X is at least 7 is not less than 0.05 (5%), which is what we need it to be.
So there is not significant evidence at the 10% level to reject the null hypothesis.
In the previous pages, you learned how to perform define the hypothesis for a statistical test, then to perform a t-test to compare means. In the example t-test we performed, we defined an alternate hypothesis to test whether one mean was greater than the other: μ > μ0.
In this situation, we tested whether one mean was higher than the other. We were not interested in whether the first mean was lower than the other, only if it was higher. So we were only interested in one side of the probability distribution, which is shown in the image below:
In this distribution, the shaded region shows the area represented by the null hypothesis, H0: μ = μ0. This actually implies μ ≤ μ0, since the only unshaded region in the image shows μ > μ0. Because we were only interested in one side of the distribution, or one "tail", this type of test is called a one-sided or a one-tailed test. When you are using tables for probability distributions, you should make sure whether they are for one-tailed or two-tailed tests. Depending on which they are for, you need to know how to switch to the one you need. This is all explained below.
A one-tailed test uses an alternate hypothesis that states either H1: μ > μ0 OR H1: μ < μ0, but not both. If you want to test both, using the alternate hypothesis H1: μ ≠ μ0, then you need to use a two-tailed test.
We would use a two-tailed test to see if two means are different from each other...