Respuesta :
Answer:
(a) At 21.46 psi, the TPMS trigger a warning for this car.
(b) The probability that the TPMS will trigger a warning is 0.0001.
(c) The probability that the tire’s inflation is within the recommended range is 0.6826.
Step-by-step explanation:
We are given that tire pressure monitoring systems (TPMS) warn the driver when the tire pressure of the vehicle is 26% below the target pressure. Suppose the target tire pressure of a certain car is 29 psi (pounds per square inch).
(a) It is stated that TPMS warns the driver when the tire pressure of the vehicle is 26% below the target pressure.
So, the TPMS trigger a warning for this car when;
Pressure = 29 psi - 26% of 29 psi
= [tex]29-(0.26 \times 29)[/tex] = 21.46 psi
At 21.46 psi, the TPMS trigger a warning for this car.
(b) Suppose tire pressure is a normally distributed random variable with a standard deviation equal to 2 psi.
Let X = The pressure at which TPMS will trigger a warning
So, X ~ Normal([tex]\mu=29, \sigma^{2} =2^{2}[/tex])
Now, the probability that the TPMS will trigger a warning is given by = P(X [tex]\leq[/tex] 21.46)
P(X [tex]\leq[/tex] 21.46) = P( [tex]\frac{X-\mu}{\sigma}[/tex] [tex]\leq[/tex] [tex]\frac{21.46-29}{2}[/tex] ) = P(Z [tex]\leq[/tex] -3.77) = 1 - P(Z < 3.77)
= 1 - 0.9999 = 0.0001
The above probability is calculated by looking at the value of x = 3.77 in the z table which has an area of 0.9999.
(c) The manufacturer’s recommended correct inflation range is 27 psi to 31 psi.
So, the probability that the tire’s inflation is within the recommended range is given by = P(27 psi < X < 31 psi)
P(27 psi < X < 31 psi) = P(X < 31 psi) - P(X [tex]\leq[/tex]27 psi)
P(X < 31 psi) = P( [tex]\frac{X-\mu}{\sigma}[/tex] < [tex]\frac{31-29}{2}[/tex] ) = P(Z < 1) = 0.8413
P(X [tex]\leq[/tex] 27 psi) = P( [tex]\frac{X-\mu}{\sigma}[/tex] [tex]\leq[/tex] [tex]\frac{27-29}{2}[/tex] ) = P(Z [tex]\leq[/tex] -1) = 1 - P(Z < 1)
= 1 - 0.8413 = 0.1587
Therefore, P(27 psi < X < 31 psi) = 0.8413 - 0.1587 = 0.6826.
At 21.46 psi, the TPMS trigger a warning for this car.
The probability that the TPMS will trigger a warning is 0.0001.
The probability that the tire’s inflation is within the recommended range is 0.6826.
Given that,
Tire pressure monitoring systems (TPMS) warn the driver when the tire pressure of the vehicle is 26% below the target pressure.
Suppose the target tire pressure of a certain car is 29 psi (pounds per square inch).
We have to determine,
At what psi will the TPMS trigger a warning for this car.
What is the probability that the TPMS will trigger a warning.
What is the probability that the tire’s inflation is within the recommended range.
According to the question,
- It is stated that TPMS warns the driver when the tire pressure of the vehicle is 26% below the target pressure.
So, the TPMS trigger a warning for this car when;
Pressure = 29 psi - 26% of 29 psi
[tex]Pressure = 29- (0.26 \times 29) = 21.46psi[/tex]
At 21.46 psi, the TPMS trigger a warning for this car.
- Suppose tire pressure is a normally distributed random variable with a standard deviation equal to 2 psi.
Let X = The pressure at which TPMS will trigger a warning
So, X ~ Normal [tex]\mu = 29, \sigma^2 = 2^2\\[/tex]
Now, the probability that the TPMS will trigger a warning is given by = P(X≤ 21.46).
[tex]P(X\leq 21.46) = P (\dfrac{X-\mu}{\sigma}\leq \dfrac{21.46-29}{2}) = P(Z\leq -3.77) = 1-P(Z<3.77) \\\\= 1-0.9999 = 0.0001[/tex]
The above probability is calculated by looking at the value of x = 3.77 in the z table which has an area of 0.9999.
- The manufacturer’s recommended correct inflation range is 27 psi to 31 psi.
So, the probability that the tire’s inflation is within the recommended range is given by = P(27 psi < X < 31 psi)
[tex]P(27psi<X<31psi) = P(X<31psi)-P(X\leq psi)\\\\P(X<31psi) = P(\dfrac{x-\mu}{\sigma}\leq \dfrac{31-29}{2}) = P(Z<1) = 0.8413\\\\P(X<27psi) = P(\dfrac{x-\mu}{\sigma}\leq \dfrac{27-29}{2}) = P(Z\leq -1) = 1-P(Z<1) = 0.8413\\[/tex]
Therefore, , P(27 psi < X < 31 psi) = 0.8413 - 0.1587 = 0.6826.
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