Explanation:
(a) To determine the psi at which the TPMS will trigger a warning, we need to find 26% of the target pressure and subtract it from the target pressure.
26% of 28 psi = 0.26 * 28 = 7.28 psi
Trigger psi = Target psi - 26% of target psi
Trigger psi = 28 psi - 7.28 psi = 20.72 psi
Therefore, the TPMS will trigger a warning when the tire pressure reaches 20.72 psi.
(b) The average tire pressure is on target, which means it is 28 psi. The standard deviation is given as 3 psi.
To find the probability that the TPMS will trigger a warning, we need to calculate the probability that the tire pressure falls below the trigger psi of 20.72 psi.
Z-score = (Trigger psi - Average psi) / Standard deviation
Z-score = (20.72 - 28) / 3 = -2.4267 (rounded to 4 decimal places)
Using a standard normal distribution table or calculator, we can find the corresponding cumulative probability for a Z-score of -2.4267, which is approximately 0.0078.
Therefore, the probability that the TPMS will trigger a warning when the average tire pressure is on target is approximately 0.0078.
(c) The tires' average psi is on target at 28 psi. We need to find the probability that a randomly inspected tire has an inflation within the recommended range of 26 psi to 30 psi.
To calculate this probability, we need to find the area under the normal distribution curve between 26 psi and 30 psi.
Z-score for 26 psi: (26 - 28) / 3 = -0.6667
Z-score for 30 psi: (30 - 28) / 3 = 0.6667
Using a standard normal distribution table or calculator, we can find the cumulative probabilities for these two Z-scores:
P(Z ≤ -0.6667) = 0.2525
P(Z ≤ 0.6667) = 0.7475
The probability that the tire's inflation is within the recommended range is the difference between these two probabilities:
P(-0.6667 ≤ Z ≤ 0.6667) = P(Z ≤ 0.6667) - P(Z ≤ -0.6667) = 0.7475 - 0.2525 = 0.4950
Therefore, the probability that a randomly inspected tire's inflation is within the recommended range is approximately 0.4950.