How to Create One-Sided Confidence Intervals (With Examples)

A confidence interval for a mean is a range of values that is likely to contain a population mean with a certain level of confidence.

It is calculated as:

Confidence Interval = x +/- t_{α/2, n-1}*(s/√n)

where:

• x: sample mean
• t_{α/2, n-1}: t-value that corresponds to α/2 with n-1 degrees of freedom
• s: sample standard deviation
• n: sample size

The formula above describes how to create a typical two-sided confidence interval.

However, in some scenarios we’re only interested in creating one-sided confidence intervals.

We can use the following formulas to do so:

Lower One-Sided Confidence Interval = [-∞, x + t_{α, n-1}*(s/√n) ]

Upper One-Sided Confidence Interval = [ x – t_{α, n-1}*(s/√n), ∞ ]

The following examples show how to create lower one-sided and upper one-sided confidence intervals in practice.

Example 1: Create a Lower One-Sided Confidence Interval

Suppose we’d like to create a lower one-sided 95% confidence interval for a population mean in which we collect the following information for a sample:

• x: 20.5
• s: 3.2
• n: 18

According to the Inverse t Distribution Calculator, the t-value that we should use for a one-sided 95% confidence interval with n-1 = 19 degrees of freedom is 1.7291.

We can then plug each of these values into the formula for a lower one-sided confidence interval:

• Lower One-Sided Confidence Interval = [-∞, x + t_{α, n-1}*(s/√n) ]
• Lower One-Sided Confidence Interval = [-∞, 20.5 + 1.7291*(3.2/√18) ]
• Lower One-Sided Confidence Interval = [-∞, 21.804 ]

We would interpret this interval as follows: We are 95% confident that the true population mean is equal to or less than 21.804.

Example 2: Create an Upper One-Sided Confidence Interval

Suppose we’d like to create an upper one-sided 95% confidence interval for a population mean in which we collect the following information for a sample:

• x: 40
• s: 6.7
• n: 25

According to the Inverse t Distribution Calculator, the t-value that we should use for a one-sided 95% confidence interval with n-1 = 24 degrees of freedom is 1.7109.

We can then plug each of these values into the formula for an upper one-sided confidence interval:

• Upper One-Sided Confidence Interval = [ x – t_{α, n-1}*(s/√n), ∞ ]
• Lower One-Sided Confidence Interval = [ 40 – 1.7109*(6.7/√25), ∞ ]
• Lower One-Sided Confidence Interval = [ 37.707, ∞ ]

We would interpret this interval as follows: We are 95% confident that the true population mean is greater than or equal to 37.707.

Additional Resources

The following tutorials provide additional information about confidence intervals:

An Introduction to Confidence Intervals
How to Report Confidence Intervals
How to Interpret a Confidence Interval that Contains Zero