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In all the following, we will use a 95% level of confidence.

Interval for one mean using Z

The formula for a confidence interval for a mean using Z is:

where Z is the critical value from a two-tail test. For example, the value of Z in a 95% confidence interval is 1.96 because P(-1.96 < Z < 1.96) = 0.95.

Example

 = 5, s = 2 and n = 30

Lower limit=  = 5 - 0.7157 = 4.2843

Upper limit = 5 + 0.7157 = 5.7157

Interval for one mean using t

The formula for a confidence interval for a mean using t is:

where t is the critical value from a two-tail test. The degrees of freedom = n – 1.

Example

 = 5, s = 2 and n = 15. Then the degrees of freedom = 14.

Lower limit = 5 - 2.145(2)/ = 5 - 1.1077 = 3.8923

Upper limit = 5 + 1.1077 = 6.1077

Interval for one proportion using Z

The formula for a confidence interval for a proportion is:

where Z is the critical value from a two-tail test.

Example

p = 54/200 = 0.27. Obviously, n = 200

Lower limit = 0.27 - 1.96 = 0.27 - 0.0615 = 0.2085

Upper limit = 0.27 + 0.0615 = 0.3315

By the way, the second term in the above formulas is known officially as the bound on the error. Usually, especially with proportions, it is called the margin of error.