### Tire Pressure Monitoring Systems (TPMS) and Warning Triggers#### Problem Statement:Tire pressure monitoring systems (TPMS) warn the driver when the tire pressure of the vehicle is 27% below the target pressure. Suppose the target tire pressure of a certain car is 31 psi (pounds per square inch).1. **Determine Warning Pressure:** - **(a) At what psi will the TPMS trigger a warning for this car?** - **Answer:** When the tire pressure is below $ \text{22.63 psi} $. (Rounded to 2 decimal places)2. **Probability of Trigger Warning:** - **(b) Suppose tire pressure is a normally distributed random variable with a standard deviation equal to 3 psi. If the car’s average tire pressure is on target, what is the probability that the TPMS will trigger a warning?** - **Answer:** - Required rounded probability (Rounded your answer to 4 decimal places)3. **Inflation Range and Probability:** - **(c) The manufacturer’s recommended correct inflation range is 29 psi to 33 psi. Assume the tires’ average psi is on target. If a tire on the car is inspected at random, what is the probability that the tire’s inflation is within the recommended range?** - **Answer:** - Required probability within the range (Round your intermediate calculations and final answer to 4 decimal places)#### Instructions for Solving:1. **Calculate the Warning Pressure:** - Find 27% of the target pressure. - Subtract this value from the target pressure.2. **Calculate the Probability of Triggering a Warning:** - Use the normal distribution properties. - Find the Z-score corresponding to 22.63 psi. - Use standard normal distribution tables or software to find the cumulative probability.3. **Calculate Probability Within the Recommended Range:** - Establish the Z-scores for 29 psi and 33 psi. - Use standard normal distribution to find the probabilities corresponding to these Z-scores. - Calculate the probability that lies between these Z-scores.#### Detailed Diagram Explanations (If Any):In case there are graphs or diagrams involved in explaining the probabilities or distributions:- **Normal Distribution Diagram:** Illustrate the bell-curve with the mean at 31 psi. - Indicate the region below 22.

a.Given, TPMS warns the driver when the fire pressure of the vehicle is 27% below the target…

Tire pressure monitoring systems (TPMS) warn the driver when the tire pressure of the vehicle is 27% below the ldi Suppose the target tire pressure of a certain car is 31 psi (pounds per square inch.) arge

Tire pressure monitoring systems (TPMS) warn the driver when the tire pressure of the vehicle is 27% below the ldi Suppose the target tire pressure of a certain car is 31 psi (pounds per square inch.) arge

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!

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### Tire Pressure Monitoring Systems (TPMS) and Warning Triggers

#### Problem Statement:
Tire pressure monitoring systems (TPMS) warn the driver when the tire pressure of the vehicle is 27% below the target pressure. Suppose the target tire pressure of a certain car is 31 psi (pounds per square inch).

1. **Determine Warning Pressure:**
- **(a) At what psi will the TPMS trigger a warning for this car?**
- **Answer:** When the tire pressure is below $ \text{22.63 psi} $. (Rounded to 2 decimal places)

2. **Probability of Trigger Warning:**
- **(b) Suppose tire pressure is a normally distributed random variable with a standard deviation equal to 3 psi. If the car’s average tire pressure is on target, what is the probability that the TPMS will trigger a warning?**
- **Answer:**
- Required rounded probability (Rounded your answer to 4 decimal places)

3. **Inflation Range and Probability:**
- **(c) The manufacturer’s recommended correct inflation range is 29 psi to 33 psi. Assume the tires’ average psi is on target. If a tire on the car is inspected at random, what is the probability that the tire’s inflation is within the recommended range?**
- **Answer:**
- Required probability within the range (Round your intermediate calculations and final answer to 4 decimal places)

#### Instructions for Solving:
1. **Calculate the Warning Pressure:**
- Find 27% of the target pressure.
- Subtract this value from the target pressure.

2. **Calculate the Probability of Triggering a Warning:**
- Use the normal distribution properties.
- Find the Z-score corresponding to 22.63 psi.
- Use standard normal distribution tables or software to find the cumulative probability.

3. **Calculate Probability Within the Recommended Range:**
- Establish the Z-scores for 29 psi and 33 psi.
- Use standard normal distribution to find the probabilities corresponding to these Z-scores.
- Calculate the probability that lies between these Z-scores.

#### Detailed Diagram Explanations (If Any):

In case there are graphs or diagrams involved in explaining the probabilities or distributions:
- **Normal Distribution Diagram:** Illustrate the bell-curve with the mean at 31 psi.
- Indicate the region below 22.

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