Confidence Interval for μ, Standard Deviation Known (1 of 3)
This section explains how to compute a confidence interval for the mean of a normally-distributed variable for which the population standard deviation is known. In practice, the population standard deviation is rarely known. However, learning how to compute a confidence interval when the standard deviation is known is an excellent introduction to how to compute a confidence interval when the standard deviation has to be estimated.
Three values are used to construct a confidence interval for μ: the sample mean (M), the value of z (which depends on the level of confidence), and the standard error of the mean (σM). The confidence interval has M for its center and extends a distance equal to the product of z and σM in both directions. Therefore, the formula for a confidence interval is:
M - z σM ≤ μ ≤ M + z σM.
Assume that the standard deviation of SAT verbal scores in a school system is known to be 100. A researcher wishes to estimate the mean SAT score and compute a 95% confidence interval from a random sample of 10 scores.