How Do Confidence Bounds Relate to a One-Tailed Test?

A one-tailed hypothesis test asks whether a parameter is higher or lower than a specified value. The confidence bound that matches the test is one-sided: a lower bound for a right-tailed test and an upper bound for a left-tailed test. Matching a test at significance level $\alpha$ generally means using a one-sided confidence level of $1-\alpha$.

The comparison is valid when the confidence bound is obtained by inverting the same test procedure. This is exact for standard one-sample $z$ and $t$ tests, for example. Other methods—such as some intervals for proportions—may not match a test unless they are constructed from that same method.

Which bound matches which test?

Suppose the parameter of interest is $\theta$ and the null boundary is $\theta_0$.

• For a right-tailed alternative, $H_1: \theta>\theta_0$, use a lower one-sided bound. Reject the null when the lower bound is greater than $\theta_0$.
• For a left-tailed alternative, $H_1: \theta<\theta_0$, use an upper one-sided bound. Reject the null when the upper bound is less than $\theta_0$.

The null is often written as $H_0: \theta\leq\theta_0$ for the right-tailed test or $H_0: \theta\geq\theta_0$ for the left-tailed test. The test’s rejection probability is evaluated at the boundary $\theta=\theta_0$.

At level $\alpha$, the corresponding one-sided confidence bound has nominal coverage $1-\alpha$. Its actual coverage is exactly $1-\alpha$ for standard exact procedures such as the one-sample $z$ and $t$ procedures under their assumptions, and may be higher for conservative procedures. Thus a 5% right-tailed test corresponds to a 95% lower confidence bound—not a 95% two-sided interval.

Formulas for a mean

For a sample mean $\bar{x}$ from a normal population with known standard deviation $\sigma$, the standard error is $\sigma/\sqrt{n}$. Let $z_{1-\alpha}$ be the $1-\alpha$ quantile of the standard normal distribution. The bounds are

Use $L$ for a right-tailed test and $U$ for a left-tailed test. If $\sigma$ is unknown and the usual one-sample $t$ assumptions apply, replace $\sigma$ with the sample standard deviation $s$ and $z_{1-\alpha}$ with the $1-\alpha$ quantile of a $t$ distribution with $n-1$ degrees of freedom.

Worked example: right-tailed test

A sample of $n=25$ measurements has mean $\bar{x}=104$. Assume the population standard deviation is known to be $\sigma=10$. Test at the 5% level whether the population mean exceeds 100:

The standard error and 95% lower confidence bound are

Because the lower bound, about 100.71, is above the null value 100, reject $H_0$ at the 5% level. The data support a mean greater than 100 under the assumed model. The equivalent test statistic is $z=(104-100)/2=2$, which gives a one-sided $p$-value of about 0.0228, also below 0.05.

For a left-tailed test using the same estimate and standard error, the relevant bound would instead be the upper bound $U=\bar{x}+z_{1-\alpha}\operatorname{SE}$. You would reject only if that upper bound were below the null value.

Why a usual two-sided interval can mislead

A 95% two-sided interval puts 2.5% of its error probability in each tail. A one-tailed test at $\alpha=0.05$ puts all 5% in the relevant tail. For a normal-based interval, the two-sided 95% interval therefore uses $z_{0.975}\approx1.96$, while the one-sided 95% bound uses $z_{0.95}\approx1.645$.

In the example, the ordinary two-sided 95% interval is approximately $[100.08,107.92]$. Its lower endpoint differs from the one-sided 95% lower bound of 100.71. In this symmetric normal setting, the one-sided 95% lower bound matches the lower endpoint of a two-sided 90% interval. This relationship depends on the interval construction; the safest approach is to use the bound obtained from the test you are performing.

Common mistakes

• Using the wrong side: A right-tailed alternative asks whether the parameter is larger, so the relevant endpoint is a lower bound. A left-tailed alternative uses an upper bound.
• Using a two-sided interval automatically: Its error probability is split between two tails, so its endpoint generally does not correspond to a one-tailed test at the same significance level.
• Comparing in the wrong direction: For a right-tailed test, reject when $L>\theta_0$. For a left-tailed test, reject when $U<\theta_0$.
• Mixing methods: A test based on a $t$ statistic should be paired with its corresponding $t$ bound. For nonstandard tests, use an interval obtained by inverting that test rather than assuming a familiar formula has the same properties.
• Treating failure to reject as proof: If the bound does not pass the null value, the test has not established the alternative. It does not prove that the null is true.