Logistic Confidence Bounds Example

This example appears in the Reliability growth reference.

For the data given in the Reliability Data - Logistic Model example above for the reliability data example, calculate the 2-sided 90% confidence bounds under the Logistic model for the following:


 * 1) The parameters $$b\,\!$$ and $$k\,\!$$.
 * 2) Reliability at month 5.

Solution  The values of $$\hat{b}\,\!$$ and $$\hat{k}\,\!$$ that were estimated from the least squares analysis in the reliability data example are:


 * $$\begin{align}

\widehat{b}= & 3.3991 \\ \widehat{\alpha }= & 0.7398 \end{align}\,\!$$

Thus, the 2-sided 90% confidence bounds on parameter $$b\,\!$$ are:


 * $$\begin{align}

{{b}_{lower}}= & 2.5547 \\ {{b}_{upper}}= & 4.5225 \end{align}\,\!$$

The 2-sided 90% confidence bounds on parameter $$k\,\!$$ are:


 * $$\begin{align}

{{k}_{lower}}= & 0.6798 \\ {{k}_{upper}}= & 0.7997 \end{align}\,\!$$

 First, calculate the reliability estimation at month 5:
 * $$\begin{align}

{{R}_{5}}= & \frac{1}{1+b{{e}^{-5k}}} \\ = & 0.9224 	\end{align}\,\!$$ Thus, the 2-sided 90% confidence bounds on reliability at month 5 are:


 * $$\begin{align}

{{[{{R}_{5}}]}_{lower}}= & 0.8493 \\ {{[{{R}_{5}}]}_{upper}}= & 0.9955 \end{align}\,\!$$

The next figure shows a graph of the reliability plotted with 2-sided 90% confidence bounds.

 