Step 1

The confidence interval for mean when the population standard deviation is known, is given by \(\displaystyle\overline{{x}}\pm{z}_{{\frac{a}{{2}}}}\frac{\sigma}{\sqrt{{n}}}\). Where \(\displaystyle\overline{{x}}\) is the sample mean which is given as 1000 hours, \(\sigma\) is population standard deviation which is given as 30 hours, n is sample size which is 15 lamps and z value depends on the confidence level and for \(90\%\) it is 1.28. For lower bound use \(\displaystyle\overline{{x}}-{z}_{\alpha}\frac{\sigma}{\sqrt{{n}}}.\)

Step 2

The 90% lower-confidence bound on the mean life is given below:

\(\displaystyle\mu\ge-{z}_{\alpha}\frac{\sigma}{\sqrt{{n}}}\)

\(\displaystyle\ge{1000}-{1.28}\frac{30}{\sqrt{{15}}}\)

\(\displaystyle\ge{1000}-{1.28}\times{7.746}\)

\(\displaystyle\ge{1000}-{9.9149}\)

\(\displaystyle\ge{990.1}\)

Thus, the \(90\%\) lower-confidence bound on the mean life is \(\displaystyle{990}\le\mu.\)

The confidence interval for mean when the population standard deviation is known, is given by \(\displaystyle\overline{{x}}\pm{z}_{{\frac{a}{{2}}}}\frac{\sigma}{\sqrt{{n}}}\). Where \(\displaystyle\overline{{x}}\) is the sample mean which is given as 1000 hours, \(\sigma\) is population standard deviation which is given as 30 hours, n is sample size which is 15 lamps and z value depends on the confidence level and for \(90\%\) it is 1.28. For lower bound use \(\displaystyle\overline{{x}}-{z}_{\alpha}\frac{\sigma}{\sqrt{{n}}}.\)

Step 2

The 90% lower-confidence bound on the mean life is given below:

\(\displaystyle\mu\ge-{z}_{\alpha}\frac{\sigma}{\sqrt{{n}}}\)

\(\displaystyle\ge{1000}-{1.28}\frac{30}{\sqrt{{15}}}\)

\(\displaystyle\ge{1000}-{1.28}\times{7.746}\)

\(\displaystyle\ge{1000}-{9.9149}\)

\(\displaystyle\ge{990.1}\)

Thus, the \(90\%\) lower-confidence bound on the mean life is \(\displaystyle{990}\le\mu.\)