remolatg

2021-02-09

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

Lacey-May Snyder

Step 1
The confidence interval for mean when the population standard deviation is known, is given by $\stackrel{―}{x}±{z}_{\frac{a}{2}}\frac{\sigma }{\sqrt{n}}$. Where $\stackrel{―}{x}$ 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 $90\mathrm{%}$ it is 1.28. For lower bound use $\stackrel{―}{x}-{z}_{\alpha }\frac{\sigma }{\sqrt{n}}.$
Step 2
The 90% lower-confidence bound on the mean life is given below:
$\mu \ge -{z}_{\alpha }\frac{\sigma }{\sqrt{n}}$
$\ge 1000-1.28\frac{30}{\sqrt{15}}$
$\ge 1000-1.28×7.746$
$\ge 1000-9.9149$
$\ge 990.1$
Thus, the $90\mathrm{%}$ lower-confidence bound on the mean life is $990\le \mu .$

Jeffrey Jordon