I was wondering given a one-sided (one-tailed) hypothesis with an alpha-level of
.05, can we be talking about 95% confidence intervals?
For example, can we construct separately “one-sided” and “two-sided” confidence intervals for a one-sided Z or t test? what would be the “interpretation” of each of these confidence intervals given the one-sided test?
I am a bit confused about this?
Yes we can construct one sided confidence intervals with 95% coverage.
The two sided confidence interval corresponds to the critical values in a two-tailed hypothesis test, the same applies to one sided confidence intervals and one-tailed hypothesis tests.
For example, if you have data with sample statistics ˉx=7, s=4 from a sample size n=40
The two-sided 95% confidence interval for the mean is 7±1.964√40=(5.76,8.24)
If we were doing a hypothesis test for μ=μ0 then the null hypothesis would be rejected if we were using a value of μ0 which is μ0>8.24 or μ0<5.76
Constructing one-sided 95% confidence intervals
In the above confidence interval we get 95% coverage with 47.5% of the population above the mean and 47.5% below the mean. In a one sided interval we can get 95% coverage with 50% below the mean and 45% above the mean.
For a standard normal distribution the value which corresponds to 50% below the mean is −∞. 45% of the population above the mean is 1.64, you can check this in any Z tables. Using the above example we get that the upper limit to the confidence interval is 7+1.644√40=8.04
The one-sided confidence interval is therefore (−∞,8.04)
If we were doing a hypothesis test for μ<μ0 then we would reject the null hypothesis if we were considering a value of μ0 that is larger than 8.04
Two sided interval for a one sided test
When you construct a two-sided 95% confidence interval (a,b) you have 2.5% of the population which is below a and 2.5% of the population is above b (hence 5% of the population is outside the interval).
You could use this for a one-sided test, if you want to test the hypothesis that μ>μ0 then check if μ0<a. If μ0<a then you reject the hypothesis μ>μ0 with a significance of 2.5%.
Do not use this to test both μ>μ0 or μ<μ0. You have to decide before you look at the data which hypothesis you are going to test. If you don't decide before then you are introducing a bias and your significance will only be 5%