# The UNIVARIATE Procedure

### Confidence Limits for Parameters of the Normal Distribution

The two-sided confidence interval for the mean has upper and lower limits

where and is the percentile of the t distribution with degrees of freedom. The one-sided upper confidence limit is computed as and the one-sided lower confidence limit is computed as . See Example 4.9.

The two-sided confidence interval for the standard deviation has lower and upper limits,

respectively, where and are the and percentiles of the chi-square distribution with degrees of freedom. A one-sided confidence limit has lower and upper limits,

respectively. The confidence interval for the variance has upper and lower limits equal to the squares of the corresponding upper and lower limits for the standard deviation.

When you use the WEIGHT statement and specify VARDEF=DF in the PROC statement, the confidence interval for the weighted mean is

where is the weighted mean, is the weighted standard deviation, is the weight for ith observation, and is the percentile for the t distribution with degrees of freedom.

Confidence intervals for the weighted standard deviation are computed by substituting for s in the preceding formulas for confidence limits for the standard deviation.