## 3.2 One-sided vs two-sided tests

In the previous section, we mentioned that the test we had carried out was a two-sided test. One-sided tests are also possible. Consider the following cases:

1. Two-sided test: Is the average cholesterol level of patients from this population different from 5.0 mmol/L?
• $$H_0:\mu = 5\;\;\text{versus}\;\;H_1:\mu \neq 5$$
2. One-sided test: Is the average cholesterol level of patients from this population greater than 5.0 mmol/L?
• $$H_0:\mu = 5\;\;\text{versus}\;\;H_1:\mu > 5$$
3. One-sided test: Is the average cholesterol level of patients from this population less than 5.0 mmol/L?
• $$H_0:\mu = 5\;\;\text{versus}\;\;H_1:\mu < 5$$

Examples 2 and 3 above are referred to as 'one-sided tests' because they are only testing for extreme values in one direction. Consider the below figure, which shows the critical values (CV) required for each test: 1. The two-sided test can also be referred to as a two-tailed test, because as we can see above, there are two tails of the distribution curve for which we are interested in extreme values. Because the combined area of the shaded area must equal $$\alpha = 0.05$$, in this case, we have an area of $$\alpha / 2 = 0.5 / 2 = 0.025$$ at each tail. This resulted in critical values of -1.99 and 1.99 respectively. For a two-sided test, we have: $p\text{-value} = 2 \times P(T\geq |t|) \text{ for } T\sim t_{\text{df}}$
2. The one-sided test (right-tailed) can also be referred to as a right-tailed test, because as we can see above, it is the right-tail of the distribution curve for which we are interested in extreme values. Because the total area of the shaded area must equal $$\alpha = 0.05$$, in this case, we simply have an area of $$\alpha = 0.5$$ in the right tail. This resulted in a critical value of 1.67. For a one-sided test (right-tailed), we have: $p\text{-value} = P(T\geq t) \text{ for } T\sim t_{\text{df}}$
3. The one-sided test (left-tailed) can also be referred to as a left-tailed test, because as we can see above, it is the left-tail of the distribution curve for which we are interested in extreme values. Because the total area of the shaded area must equal $$\alpha = 0.05$$, in this case, we simply have an area of $$\alpha = 0.5$$ in the left tail. This resulted in a critical value of -1.67. For a one-sided test (left-tailed), we have: $p\text{-value} = P(T\leq t) \text{ for } T\sim t_{\text{df}}$

Two-sided tests are often preferred in practice because they are unbiased in terms of the predicted direction of the results. However, in this subject we will get practice using both two-sided and one-sided tests.

1. $$H_0:\mu = 4\;\;\text{versus}\;\;H_1:\mu > 4$$
2. $$H_0:\mu = 4\;\;\text{versus}\;\;H_1:\mu \neq 4$$
3. $$H_0:\mu = 4\;\;\text{versus}\;\;H_1:\mu < 4$$