The speeds of vehicles on a highway with speed limit 110 km/h are normally distributed with mean 125 km/h and standard deviation 5 km/h. (Round your answers to two decimal places.)
(a) What is the probability that a randomly chosen vehicle is traveling at a legal speed?
 %

(b) If police are instructed to ticket motorists driving 130 km/h or more, what percentage of motorist are targeted?

To solve these problems, we use the properties of the normal distribution and the Z-score formula. 

Given:
- Mean (\(\mu\)) = 125 km/h
- Standard deviation (\(\sigma\)) = 5 km/h

### Part (a) - Probability of Traveling at a Legal Speed

A legal speed is any speed less than or equal to 110 km/h.

1. **Calculate the Z-score for 110 km/h**:
   \[
   Z = \frac{X - \mu}{\sigma}
   \]
   where \(X\) is the value of interest (110 km/h).

   \[
   Z = \frac{110 - 125}{5} = \frac{-15}{5} = -3
   \]

2. **Find the probability corresponding to this Z-score**:
   Using the standard normal distribution table, or a calculator, the probability of \(Z \leq -3\) is approximately 0.00135.

   Therefore, the probability that a randomly chosen vehicle is traveling at a legal speed is 0.135%.

### Part (b) - Percentage of Motorists Targeted for Speeds 130 km/h or More

1. **Calculate the Z-score for 130 km/h**:
   \[
   Z = \frac{130 - 125}{5} = \frac{5}{5} = 1
   \]

2. **Find the probability corresponding to this Z-score**:
   Using the standard normal distribution table, the probability of \(Z \leq 1\) is approximately 0.8413.

3. **Calculate the probability for speeds 130 km/h or more**:
   Since the normal distribution table gives us the probability of being less than a given value, we subtract this value from 1.

   \[
   P(X \geq 130) = 1 - P(X \leq 130) = 1 - 0.8413 = 0.1587
   \]

   Therefore, approximately 15.87% of motorists are targeted for driving at speeds of 130 km/h or more.

### Summary
- (a) The probability that a randomly chosen vehicle is traveling at a legal speed is approximately **0.14%**.
- (b) The percentage of motorists targeted for driving 130 km/h or more is approximately **15.87%**.

Do you want details or have any questions?

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1. What is a normal distribution?
2. How do you calculate a Z-score?
3. What is the importance of the mean in a normal distribution?
4. Why is the standard deviation important in statistics?
5. How do you interpret the Z-score value?
6. What is the standard normal distribution table?
7. How can the probability value be used in real-life scenarios?
8. What are some other examples of normally distributed data in real life?

**Tip:** Understanding how to read and interpret the Z-score from the standard normal distribution table is crucial for solving problems involving normal distributions.

Learn how to calculate probabilities and Z-scores in a normal distribution scenario where vehicle speeds follow a specified mean and standard deviation. Solve for legal speed probability and target percentage for speeding motorists. Suitable for high school students studying statistics and probability.

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