The UNIVARIATE Procedure

Confidence Limits for Parameters of the Normal Distribution

The two-sided $100(1-\alpha )\%$ confidence interval for the mean has upper and lower limits

$\bar{x} \pm t_{1-\frac{\alpha }{2};n-1}\frac{s}{\sqrt {n}}$

where $s^2 = \frac{1}{n-1}\sum (x_ i -\bar{x})^2$ and $t_{1-\frac{\alpha }{2};n-1}$ is the $(1-\frac{\alpha }{2})$ percentile of the $t$ distribution with $n-1$ degrees of freedom. The one-sided upper $100(1-\alpha )\%$ confidence limit is computed as $\bar{x} + \frac{s}{\sqrt {n}} t_{1-\alpha ;n-1}$ and the one-sided lower $100(1-\alpha )\%$ confidence limit is computed as $\bar{x} - \frac{s}{\sqrt {n}} t_{1-\alpha ;n-1}$ . See Example 4.9.

The two-sided $100(1-\alpha )\%$ confidence interval for the standard deviation has lower and upper limits,

$\begin{array}{ccc} s \sqrt {\frac{n-1}{\chi ^2_{1-\frac{\alpha }{2};n-1}}} & \mbox{and} & s \sqrt {\frac{n-1}{\chi ^2_{\frac{\alpha }{2};n-1}}} \end{array}$

respectively, where $\chi ^2_{1-\frac{\alpha }{2};n-1}$ and $\chi ^2_{\frac{\alpha }{2};n-1}$ are the $(1-\frac{\alpha }{2})$ and $\frac{\alpha }{2}$ percentiles of the chi-square distribution with $n-1$ degrees of freedom. A one-sided $100(1-\alpha )\%$ confidence limit has lower and upper limits,

$\begin{array}{ccc} s \sqrt {\frac{n-1}{\chi ^2_{1-\alpha ;n-1}}} & \mbox{and} & s \sqrt {\frac{n-1}{\chi ^2_{\alpha ;n-1}}} \end{array}$

respectively. The $100(1-\alpha )\%$ 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 $100(1-\alpha )\%$ confidence interval for the weighted mean is

$\bar{x}_ w \pm t_{1-\frac{\alpha }{2}} \frac{s_ w}{\sqrt {\sum _{i=1}^ n w_ i}}$

where $\bar{x}_ w$ is the weighted mean, $s_ w$ is the weighted standard deviation, $w_ i$ is the weight for $i$ th observation, and $t_{1-\frac{\alpha }{2}}$ is the $(1-\frac{\alpha }{2})$ percentile for the $t$ distribution with $n-1$ degrees of freedom.

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