Question

The lifespan of a 100-W fluorescent lamp is define to be normally distributed with displaystylesigma={30} hrs. A random sample of 15 lamps has a mean life of displaystyle{x}={1000} hours. Construct a 90% lower-confidence bound on the mean life.

Confidence intervals
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asked 2021-02-09
The lifespan of a 100-W fluorescent lamp is define to be normally distributed with \(\displaystyle\sigma={30}\) hrs. A random sample of 15 lamps has a mean life of \(\displaystyle{x}={1000}\) hours.
Construct a \(90\%\) lower-confidence bound on the mean life.

Answers (1)

2021-02-10
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 \(\displaystyle{990}\le\mu.\)
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