Problem workshop 7

It is vital that an area be lighted continuously

• they use 3 LED bulbs with 15k hrs of lifetime

  • $sd=1500\text{hrs}$

• only one bulb is needed to achieve sufficient lighting

• with a circuit that will turn on the next bulb in line when one fails

  • What is theprobability it will last for 50k hrs

$$\begin{array}{r}\text{the expected mean lifetime of the lighting will be}\\ {\mu }_{total}={\mu }_1+{\mu }_2+{\mu }_3=15000+15000+15000=45000\\ \text{standard deviation}:\\ {\sigma }_{total}=\sqrt{{\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2}=2600hrs\\ \text{this doesnt give significant so}\alpha =0.05,\text{ie 95 conifidence}\\ \text{this is a upper tail hypothesis test}\\ H_0:\mu =50000\text{hrs null}\\ H_1:\mu <50000\text{hrs alternate}\\ z_1=\frac{x-\mu }{\sigma }=\frac{45000-50000}{2600}=-1.92\\ P(z_1)=\frac{1}{2}\left[1+erf\left(-\frac{1.92}{\sqrt{2}}\right)\right]=0.019\\ \text{0.019<<}\alpha \text{ so must reject the }{H}_0\text{ hypothesis, thus only 1.9}\end{array}$$