Assume The Speed Of Vehicles Along A Stretch Of Road ✓ Solved
Assume The Speed Of Vehicles Along A Stretch Of
Assume the speed of vehicles along a stretch of I-10 has an approximately normal distribution with a mean of 71 mph and a standard deviation of 8 mph.
a. The current speed limit is 65 mph. What is the proportion of vehicles less than or equal to the speed limit?
b. What proportion of the vehicles would be going less than 50 mph?
c. A new speed limit will be initiated such that approximately 10% of vehicles will be over the speed limit. What is the new speed limit based on this criterion?
d. In what way do you think the actual distribution of speeds differs from a normal distribution?
Sample Paper For Above instruction
The analysis of vehicle speeds along a stretch of highway provides valuable insights into driver behavior and traffic safety. Assuming that the speeds follow an approximately normal distribution with a mean of 71 mph and a standard deviation of 8 mph, we can calculate various probabilities and determine appropriate speed limits based on statistical methods.
Part A: Proportion of Vehicles at or Below the Current Speed Limit
The current speed limit is 65 mph. To find the proportion of vehicles traveling at or below this speed, we standardize the value using the z-score formula:
z = (X - μ) / σ
Where:
- X = 65 mph
- μ = 71 mph
- σ = 8 mph
Calculating the z-score:
z = (65 - 71) / 8 = -6 / 8 = -0.75
Using standard normal distribution tables or a calculator, the cumulative probability corresponding to z = -0.75 is approximately 0.2266. Therefore, about 22.66% of vehicles are traveling at or below the current speed limit of 65 mph.
Part B: Proportion of Vehicles Traveling Less Than 50 mph
Standardize the value of 50 mph:
z = (50 - 71) / 8 = -21 / 8 = -2.625
The cumulative probability for z = -2.625 is approximately 0.0044, indicating that roughly 0.44% of vehicles travel at speeds less than 50 mph. This suggests that very few drivers are significantly below the average speed.
Part C: Determining a New Speed Limit for the Top 10%
To find the speed limit that only 10% of vehicles exceed, we identify the 90th percentile of the distribution. The z-score corresponding to the 90th percentile is approximately 1.28.
Calculate the corresponding speed:
X = μ + zσ = 71 + (1.28)(8) = 71 + 10.24 = 81.24 mph
Therefore, setting the new speed limit at approximately 81 mph ensures that only about 10% of vehicles will be over the limit.
Part D: How Actual Speeds May Differ from a Normal Distribution
While assuming a normal distribution facilitates calculation, actual vehicle speeds may deviate due to various factors such as traffic congestion, road conditions, or driver behavior. Empirical data often reveal skewness, kurtosis, or multiple modes, indicating that the distribution might be slightly skewed or have heavier tails than a perfect normal distribution. Additionally, external influences like speed enforcement or weather conditions can create discrepancies, leading to real-world distributions that are asymmetrical or have irregular peaks not captured by the normal model.
References
- Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (8th ed.). Cengage Learning.
- Johnson, N. L., & Kotz, S. (1970). Distributions in Statistics: Continuous Univariate Distributions. Wiley.
- Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics (9th ed.). W. H. Freeman.
- Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis (6th ed.). Brooks Cole.
- Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
- Gelman, A., et al. (2013). Bayesian Data Analysis (3rd ed.). CRC Press.
- Freeman, S., et al. (2014). Biological Science (4th ed.). Pearson.
- Larson, R., et al. (2017). Elementary Statistics (7th ed.). Pearson.
- Newbold, P., et al. (2017). Statistics for Business and Economics (9th ed.). Pearson.
- Rice, J. A. (2006). Mathematical Statistics and Data Analysis (3rd ed.). Cengage Learning.