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Creating one-sided confidence intervals involves calculating an upper or lower confidence interval bound for a parameter of interest in a population with a given level of confidence. This is done by using an appropriate z-score for the confidence level, and then subtracting or adding this value to the parameter estimate, depending on whether an upper or lower confidence interval is required. Examples are given to demonstrate the process.
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 +/- tα/2, n-1*(s/√n)
where:
- x: sample mean
- tα/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 + tα, n-1*(s/√n) ]
Upper One-Sided Confidence Interval = [ x – tα, 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 , the t-value that we should use for a one-sided 95% confidence interval with n-1 = 17 degrees of freedom is 1.7396.
We can then plug each of these values into the formula for a lower one-sided confidence interval:
- Lower One-Sided Confidence Interval = [-∞, x + tα, n-1*(s/√n) ]
- Lower One-Sided Confidence Interval = [-∞, 20.5 + 1.7396*(3.2/√18) ]
- Lower One-Sided Confidence Interval = [-∞, 21.812 ]
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 , 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 – tα, 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.
The following tutorials provide additional information about confidence intervals: