Question

Question 5: Lightbulb lifetimes follow an unknown distribution, with a mean length of 10780 hours and a standard deviation of 370 hours. Suppose that a sample of 61 lightbulbs is taken. What is the probability that the total lifetime of these lightbulbs is less than 656193 hours?

Answer 1

As a sample of 61 bulbs is taken we can use the central limit theorem and this can be approximated by a normal distribution.

Here let

X:Liftime of a bulb

$X \sim N(10780,370^2)$

To find

$P(\sum_{1}^{61}Xi <656193)$

Now using the property of normal distribution

$\sum_{1}^{61}Xi \sim N(61*10780,61*370^2)$

So

$P(\sum_{1}^{61}Xi <656193)=P(\sum_{1}^{61}Xi-(61*10780)/\sqrt{61}*370)<656193-(61*10780)/\sqrt{61}*370))$

$=P(z<-1387/\sqrt{61}*370)=P(z<-0.479965)$

$=0.3156261 \approx 0.316$

So the probability is 0.316.

Do comment if you have any doubt.

Thank you !!