*86*

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 _{Î±/2, n-1}*(s/âˆšn)**

where:

**x:Â**sample mean**t**t-value that corresponds to Î±/2 with n-1 degrees of freedom_{Î±/2, n-1}:**s:Â**sample standard deviation**n:Â**sample size

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

However, in some scenarios weâ€™re only interested in creating **one-sided confidence intervals**.

We can use the following formulas to do so:

**Lower One-Sided Confidence Interval** = [-âˆž, x + t_{Î±, n-1}*(s/âˆšn) ]

**Upper One-Sided Confidence Interval** = [ x â€“ t_{Î±, n-1}*(s/âˆšn), âˆž ]

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 weâ€™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:Â**20.5**s:Â**3.2**n:Â**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**= [-âˆž, x + t_{Î±, n-1}*(s/âˆšn) ]**Lower One-Sided Confidence Interval**= [-âˆž, 20.5 + 1.7291*(3.2/âˆš18) ]**Lower One-Sided Confidence Interval**= [-âˆž, 21.804Â ]

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 weâ€™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:Â**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 â€“ t_{Î±, n-1}*(s/âˆšn), âˆž ]**Lower One-Sided Confidence Interval**= [ 40 â€“ 1.7109*(6.7/âˆš25), âˆž ]**Lower One-Sided Confidence Interval**= [ 37.707, âˆž ]

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