Normal Distribution: Speed Of Cars

Normal Distribution1 Suppose That The Speed At Which Cars Go On The F

Suppose that the speed at which cars travel on the freeway follows a normal distribution with a mean of 69 mph and a standard deviation of 8 mph. Let X be the speed of a randomly selected car. Using this information, answer the following questions:

a. What is the distribution of X?

b. If one car is randomly chosen, what is the probability that it is traveling more than 68 mph?

c. If one of the cars is randomly chosen, what is the probability that it is traveling between 72 and 76 mph?

Paper For Above instruction

Introduction

The application of normal distribution in real-world scenarios, such as vehicle speeds on a highway, provides valuable insights into traffic flow and safety measures. By understanding the distribution parameters and utilizing standard normal calculations, we can estimate the probability of certain speed ranges, which aids in traffic management and policy-making.

Distribution of the Vehicle Speeds

The problem states that the speeds of vehicles on the freeway are normally distributed with a mean (μ) of 69 mph and a standard deviation (σ) of 8 mph. Therefore, the distribution of the vehicle speeds, X, can be expressed as:

X ~ N(69, 8^2)

This notation indicates that the random variable X follows a normal distribution with mean 69 and variance 64.

Calculating Probabilities Using the Normal Distribution

To compute the probabilities, we standardize using the Z-score transformation:

Z = (X - μ) / σ

where Z follows a standard normal distribution, N(0,1). These calculations enable us to find the probabilities for various speed ranges.

Part b: Probability of Traveling More Than 68 mph

We need to find P(X > 68). First, calculate the Z-score for X = 68:

Z = (68 - 69) / 8 = -1 / 8 = -0.125

Using standard normal tables or a calculator, find P(Z > -0.125):

P(Z > -0.125) = 1 - P(Z ≤ -0.125) = 1 - 0.4503 = 0.5497

Therefore, the probability that a randomly chosen car is traveling faster than 68 mph is approximately 0.5497.

Part c: Probability of Traveling Between 72 and 76 mph

Calculate the Z-scores for X = 72 and X = 76:

Z for 72 = (72 - 69) / 8 = 3 / 8 = 0.375

Z for 76 = (76 - 69) / 8 = 7 / 8 = 0.875

Next, find P(0.375 ≤ Z ≤ 0.875):

P(Z ≤ 0.875) = 0.8092

P(Z ≤ 0.375) = 0.6462

Thus, P(72 ≤ X ≤ 76) = 0.8092 - 0.6462 = 0.1630

In conclusion, the probability that a randomly selected vehicle travels between 72 and 76 mph is approximately 0.1630.

Conclusion

The normal distribution model effectively evaluates the likelihood of various vehicle speeds, enabling better traffic analysis and safety monitoring. The calculations show that slight deviations from the mean significantly influence the probabilities, emphasizing the importance of understanding the distribution parameters.

References

  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
  • Glantz, S. A. (2012). Primer of Biostatistics. McGraw-Hill Education.
  • Wackerly, D., Mendenhall, W., & Scheaffer, R. (2008). Mathematical Statistics with Applications. Cengage Learning.
  • Rice, J. A. (2006). Mathematical Statistics and Data Analysis. Cengage Learning.
  • Moore, D. S., Notz, W. I., & Fligner, M. A. (2013). The Basic Practice of Statistics. W.H. Freeman.
  • Newbold, P., Carlson, W., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
  • Blitzstein, J., & Hwang, J. (2014). Introduction to Probability. CRC Press.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury.
  • Lehmann, E. L., & Casella, G. (1998). Theory of Point Estimation. Springer.
  • Johnson, N. L., & Kotz, S. (1970). Distributions in Statistics: Continuous Univariate Distributions. Wiley.

Related Assignments