Question

Let X denote the time in hours needed to locate and correct a problem in the software that governs the timing of traffic lights in the downtown area of a large city. Assume that X is normally distributed with mean 10 hours and variance 9.

a. Find the probability that the next problem will require at most 15 hours to find and correct.

b. The fastest 5% of repairs take at most how many hours to complete?

Answer 1

$a:\\P\left( X\leqslant 15 \right) =P\left( \frac{X-10}{\sqrt{9}}\leqslant \frac{15-10}{\sqrt{9}} \right) =P\left( Z\leqslant 1.6667 \right) =\varPhi \left( 1.6667 \right) =0.952\\b:\\P\left( X\leqslant C \right) =0.05\Rightarrow P\left( \frac{X-10}{\sqrt{9}}\leqslant \frac{C-10}{\sqrt{9}} \right) =0.05\Rightarrow \\\Rightarrow P\left( Z\leqslant \frac{C-10}{3} \right) =0.05\Rightarrow \frac{C-10}{3}=z_{0.05}\Rightarrow C=3z_{0.05}+10=3\cdot \left( -1.6449 \right) +10=5.0653$