Question

Tire pressure monitoring systems (TPMS) warn the driver when the tire pressure of the vehicle is 28% below the target pressure. Suppose the target tire pressure of a certain car is 30 psi (pounds per square inch.)
(a) At what psi will the TPMS trigger a warning for this car?
(b) Suppose the tire pressure is a normally distributed random variable with a standard deviation equal to 2psi. If the car's average tire pressure is on target, what is the probability that the TPMS will trigger a warning?

Answer 1

Answer:

$2.95\times10^{-5}$

Step-by-step explanation:

a)The pressure level that will trigger a warning is

30 - 30*0.28 = 21.6 psi

b) The probability that the TPMS will trigger warning at 21.6 psi, given that tire pressure has a normal distribution with average of 30 psi and standard deviation of 2 psi is:

$f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}$

where x = 21.6, μ = 30 and σ = 2

$f(22.94)={\frac {1}{2 {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {21.6-30}{2 }}\right)^{2}}$

$f(22.94)=0.2e^{-8.82} = 2.95\times10^{-5}$