A *one-tailed test* results from an alternative hypothesis which specifies a direction. i.e. when the alternative hypothesis states that the parameter is in fact either bigger or smaller than the value specified in the null hypothesis.

A *two-tailed test* results from an alternative hypothesis which does not specify a direction. i.e. when the alternative hypothesis states that the null hypothesis is wrong.

A one-tailed test may be either left-tailed or right-tailed.

A *left-tailed* test is used when the alternative hypothesis states that the true value of the parameter specified in the null hypothesis is less than the null hypothesis claims.

A *right-tailed* test is used when the alternative hypothesis states that the true value of the parameter specified in the null hypothesis is greater than the null hypothesis claims

The main difference between one-tailed and two-tailed tests is that one-tailed tests will only have one critical region whereas two-tailed tests will have two critical regions. If we require a $100(1-\alpha)$% confidence interval we have to make some adjustments when using a two-tailed test.

The confidence interval must remain a constant size, so if we are performing a two-tailed test, as there are twice as many critical regions then these critical regions must be half the size. This means that when we read the tables, when performing a two-tailed test, we need to consider $\frac{\alpha}{2}$ rather than $\alpha$.

A light bulb manufacturer claims that its' energy saving light bulbs last an average of 60 days. Set up a hypothesis test to check this claim and comment on what sort of test we need to use.

So we have

- $H_0$ : The mean lifetime of an energy-saving light bulb is $60$ days.
- $H_1$ : The mean lifetime of an energy-saving light bulb is not $60$ days.

Because of the “is not” in the alternative hypothesis, we have to consider both the possibility that the lifetime of the energy-saving light bulb is greater than $60$ and that it is less than $60$. This means we have to use a two-tailed test.

The manufacturer now decides that it is only interested whether the mean lifetime of an energy-saving light bulb is less than 60 days. What changes would you make from Example 1?

So we have

- $H_0$ : The mean lifetime of an energy-saving light bulb is $60$ days.
- $H_1$ : The mean lifetime of an energy-saving light bulb is less than $60$ days.

Now we have a “less than” in the alternative hypothesis. This means that instead of performing a two-tailed test, we will perform a left-sided one-tailed test.

Find the critical values of the normal distribution using a $5$% significance level for both a one-tailed and a two-tailed test.

For the (right-sided) one-tailed test with a $5$% significance level, $z_{1-\alpha}=1.645$. (A left-tailed test would result in the critical value of $-1.645$.) For the two tailed test, $z_{1-\frac{\alpha}{2} } = 1.96$. The values are obtained from the tables of the inverse of the cumulative distribution function of the normal distribution.