Final answer:
The probability that a randomly chosen vehicle is traveling at a legal speed is 13.28%. About 13.28% of motorists traveling at 120 km/h or more are targeted for tickets, according to the normal distribution of speeds with a mean of 110 km/h and a standard deviation of 9 km/h.
Step-by-step explanation:
To answer the student's questions about the probability of vehicles traveling at legal speeds on a highway and the percentage of motorists that may be ticketed for exceeding a certain speed, we'll apply concepts from statistics, specifically the normal distribution and z-scores.
Probability of Traveling at Legal Speed
The legal speed is 100 km/h or less. To find the probability that a vehicle is traveling at or below this speed, we'll calculate the z-score using the formula:
Z = (X - μ) / σ
Where X is the speed limit (100 km/h), μ is the mean speed (110 km/h), and σ is the standard deviation (9 km/h). The z-score tells us how many standard deviations below the mean our value is:
Z = (100 - 110) / 9 ≈ -1.11
Using a standard normal distribution table, we find that the probability of a z-score being less than -1.11 is approximately 0.13, or 13.28%.
Percentage of Motorists Ticketed
Next, to find the percentage of motorists who are driving at 120 km/h or more and will be ticketed, we need the z-score for 120 km/h:
Z = (120 - 110) / 9 ≈ 1.11
The probability of a z-score being less than 1.11 is about 0.87, or 86.72%. To find the percentage above this, we subtract from 1:
1 - 0.87 = 0.13, or 13.28%
Therefore, about 13.28% of the motorists are targeted for tickets.