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# 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 +/- tÎ±/2, n-1*(s/âˆšn)

where:

• x:Â sample mean
• tÎ±/2, n-1: t-value that corresponds to Î±/2 with n-1 degrees of freedom
• 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 + tÎ±, n-1*(s/âˆšn) ]

Upper One-Sided Confidence Interval = [ x â€“ tÎ±, 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 + tÎ±, 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€“ tÎ±, 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.