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
ANSWERED 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. 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 $$\displaystyle{990}\le\mu.$$