Your solution using the GRASS format is mostly correct, but it can be improved with clearer formatting and a slight adjustment for precision in some of the calculations. Here’s an improved version:

G – Given:

• Mean speed (μ): 48 km/h
• Standard deviation (σ): 6 km/h
• Speed limit: 40 km/h
• Drivers are ticketed if they exceed the speed limit by more than 10%.

R – Required:

Find the percentage of drivers who will receive a ticket. This means determining the probability of a driver exceeding a speed of 44 km/h.

A – Approach:

1. Determine the threshold speed for ticketing:

   • The speed limit is 40 km/h.
   • 10% over the speed limit is calculated as:
     $40 \times 1.1 = 44 \, \text{km/h}$
   • Thus, drivers will be ticketed if they exceed 44 km/h.

2. Find the z-score for 44 km/h:
   The z-score formula is:
   $z = \frac{x - \mu}{\sigma}$
   Where:

   • $x$ = 44 km/h (threshold speed)
   • $\mu$ = 48 km/h (mean speed)
   • $\sigma$ = 6 km/h (standard deviation)

3. Calculate the z-score:
   $z = \frac{44 - 48}{6} = \frac{-4}{6} = -0.67$

4. Find the probability for the z-score:

   • A z-score of -0.67 corresponds to a cumulative probability of approximately 0.2514. This means that about 25.14% of drivers are driving at or below 44 km/h.

5. Find the percentage of drivers exceeding 44 km/h:

   • To find the percentage of drivers exceeding 44 km/h, subtract this cumulative probability from 1:
     $P(\text{drivers exceeding 44 km/h}) = 1 - 0.2514 = 0.7486$
   • Thus, approximately 74.86% of drivers are expected to receive a ticket.

S – Solution:

Approximately 74.86% of drivers will receive a ticket based on the speed camera's threshold.

S – Statement:

Based on the analysis, approximately 74.86% of drivers will exceed the threshold speed of 44 km/h and therefore will receive a ticket.

Improvements Made:

1. Clarity: Improved the structure and flow for easier readability.
2. Formula Presentation: Added LaTeX-like notation for better understanding of formulas.
3. Explanation: Clarified the reasoning for each step to make it more educational.
4. Precision: Made sure the steps and values are explained clearly for accuracy.

Would you like to go deeper into any of these steps or have any more questions?

5 Related Questions:

1. What is the z-score and how does it relate to probability in normal distributions?
2. How do you interpret a z-score value in practical situations?
3. What would the outcome be if the mean speed were 50 km/h instead?
4. How does changing the standard deviation affect the percentage of ticketed drivers?
5. How can you use a graphing calculator to find probabilities associated with z-scores?

Tip:

When solving normal distribution problems, sketching a bell curve and marking key values can help visualize the area under the curve and make understanding probabilities easier.