Continuous Random Variables - Normal Probabilities and Quantiles

IB AA SL: core and optional content

Core: single-bulb interval and binomial box rejection. Optional: difference involving the average fluorescent-bulb lifespan.

The lifespan of a halogen bulb is normally distributed with mean \(160\) hours with standard deviation \(10\) hours, while the lifespan of a fluorescent bulb is normally distributed with mean \(240\) hours and standard deviation \(12\) hours. The lifespan of any bulb is independent of one another.

1. A halogen bulb is randomly chosen. Find the greatest value of \(a\), correct to three significant figures, if the probability that its lifespan lies in the range of \((160-a,160+a)\) is at most \(0.4\). [3]
2. Find the probability that the difference between the average lifespan of two fluorescent bulbs and twice the lifespan of a halogen bulb does not exceed \(70\) hours. [3]
3. The halogen bulbs are packed in boxes of \(n\) bulbs. If there are more than \(10\) bulbs that have lifespans of less than \(150\) hours, the box will be rejected. Find the greatest value of \(n\) so that the probability that a box will be rejected is less than \(0.2\). [5]