Answer to Question #246496 in Statistics and Probability for sereia

Question

Leo a working student from STEM, rides a bicycle from house to school to save money. If the mean life of a tire is 10,000 km and the standard deviation is 200 km. 68% of all tires will have a life between how many kilometers?

Answer 1

Let X be a random variable representing the life of a tire. Then,

$X$~$N(\mu,\sigma^2)$ where,

$\mu=10,000$

$\sigma^2=200^2=40,000$

Hence, $X$~$(10000, 40000)$

According to empirical rule, the area covered between mean minus 1 standard deviation and mean plus 1 standard deviation of a Normal distribution is 68%. This can be written as,

$p(\mu-\sigma\lt X \lt \mu+\sigma)=68\%$

$p(10000-200 \lt X \lt 10000+200)=68\%$

$p(9800 \lt X \lt 10200)=68\%$

Therefore, $68\%$ of all tires will have a life between 9800km and 10200km.