Question

8. Tires become illegal when their tread depths fall below a certain value. A particular brand and model of tire has lifespans (number of miles that can be driven before they become illegal) that are approximately normal in distribution with mean 34,000 miles and standard deviation 3,800 miles. Use this information (and tables as needed) to answer parts a-c below. a) Determine the probability that a single tire, selected at random, has a lifespan of between 30,000 and 40,000 miles. Show the work. [4] 8b) How many miles would a tire of this brand and model have to last in order for it to be in the top 5% of all tire lifespans? [3] 8c) If a sample of 50 of these tires were collected, what is the probability that the sample mean lifespan would be less than 32,500 miles?

Answer 1

$(a)~Let~X=lifespans~of~a~tire\sim N(34000,3800^2).\\ Required~probability=P(30000\leq X\leq 40000)\\ =P\left(\frac{30000-34000}{3800}\leq \frac{X-34000}{3800}\leq \frac{40000-34000}{3800} \right )\\ =\Phi(1.5789)-\Phi( -1.0526)=0.7966\\ (b)~P(X>c)=0.05\Rightarrow \Phi((c-34000)/3800)=\Phi(1.6449)\\ or,~c=34000&plus;1.6449*3800=40250.62\approx 40251~miles\\ (c)~\bar{X}=\frac{1}{50}\sum_{i=1}^{50}X_i\sim N(34000,3800^2/50)\\ P(\bar{X}<32500)=\Phi(\sqrt{50}(32500-34000)/3800)=1-\Phi(2.7912)\\ =0.0026$