Terri Vogel, An Amateur Motorcycle Racer, Averages 129.72 Seconds Per 2.5-mile Lap (in A 7-lap Race) With A Standard Deviation Of 2.3 Seconds. The Distribution Of Her Race Times Is Normally Distributed. We Are Interested In One Of Her Randomly Selected
Introduction
In the world of sports, timing is everything. For amateur motorcycle racer Terri Vogel, understanding her average lap times and the distribution of her race times can be crucial in improving her performance. In this article, we will delve into the world of normal distributions and explore how Terri Vogel's race times can be analyzed using statistical methods.
The Normal Distribution
The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In the context of Terri Vogel's motorcycle racing times, the normal distribution can be used to model the probability of her lap times.
Terri Vogel's Lap Times
Terri Vogel averages 129.72 seconds per 2.5-mile lap in a 7-lap race, with a standard deviation of 2.3 seconds. The distribution of her race times is normally distributed. To better understand her lap times, we can use the 68-95-99.7 rule, which states that about 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and about 99.7% falls within three standard deviations.
Calculating the Z-Score
To calculate the z-score, we can use the following formula:
z = (X - μ) / σ
where X is the value we want to find the z-score for, μ is the mean, and σ is the standard deviation.
Let's say we want to find the z-score for a lap time of 130 seconds.
z = (130 - 129.72) / 2.3 z = 0.28 / 2.3 z = 0.122
Finding the Probability
Using a standard normal distribution table or calculator, we can find the probability that Terri Vogel's lap time is less than 130 seconds.
P(X < 130) = P(Z < 0.122) = 0.5506
This means that there is a 55.06% chance that Terri Vogel's lap time will be less than 130 seconds.
Finding the Interval
We can also use the z-score to find the interval within which 95% of the data falls.
P(129.72 - 2.3 < X < 129.72 + 2.3) = P(-0.58 < Z < 0.58) = 0.7194
This means that there is a 71.94% chance that Terri Vogel's lap time will fall within the interval of 127.42 to 132.02 seconds.
Conclusion
In conclusion, the normal distribution can be used to model the probability of Terri Vogel's lap times. By calculating the z-score and finding the probability, we can gain a better understanding of her lap times and make predictions about her performance. The 68-95-99.7 rule can also be used to estimate the interval within which 95% of the data falls.
Future Research Directions
There are several future research directions that can be explored in this area. Some possible research questions include:
- How does Terri Vogel's lap time distribution change over time?
- What are the factors that affect Terri Vogel's lap time distribution?
- Can we use machine learning algorithms to predict Terri Vogel's lap times?
References
- [1] Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
- [2] Johnson, R. A., & Bhattacharyya, G. K. (2014). Statistics: Principles and methods. John Wiley & Sons.
Appendix
Calculating the Standard Deviation
To calculate the standard deviation, we can use the following formula:
σ = √(Σ(xi - μ)^2 / (n - 1))
where xi is each data point, μ is the mean, and n is the number of data points.
Let's say we have the following data points:
| Lap Time | 128.5 | 129.2 | 130.1 | 131.5 | 132.2 | 133.1 | 134.5 |
|---|
We can calculate the standard deviation as follows:
σ = √(Σ(xi - 129.72)^2 / (7 - 1)) = √((128.5 - 129.72)^2 + (129.2 - 129.72)^2 + (130.1 - 129.72)^2 + (131.5 - 129.72)^2 + (132.2 - 129.72)^2 + (133.1 - 129.72)^2 + (134.5 - 129.72)^2) / 6 = √(1.22^2 + 0.52^2 + 0.38^2 + 1.78^2 + 2.48^2 + 3.38^2 + 4.78^2) / 6 = √(1.50 + 0.27 + 0.15 + 3.17 + 6.17 + 11.35 + 22.95) / 6 = √45.46 / 6 = √7.57 = 2.75
Q: What is the normal distribution?
A: The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.
Q: How does the normal distribution apply to Terri Vogel's lap times?
A: The normal distribution can be used to model the probability of Terri Vogel's lap times. By calculating the z-score and finding the probability, we can gain a better understanding of her lap times and make predictions about her performance.
Q: What is the z-score?
A: The z-score is a measure of how many standard deviations an observation is away from the mean. It can be calculated using the following formula:
z = (X - μ) / σ
where X is the value we want to find the z-score for, μ is the mean, and σ is the standard deviation.
Q: How do I calculate the z-score?
A: To calculate the z-score, you can use a calculator or a standard normal distribution table. For example, if we want to find the z-score for a lap time of 130 seconds, we can use the following formula:
z = (130 - 129.72) / 2.3 z = 0.28 / 2.3 z = 0.122
Q: What is the probability that Terri Vogel's lap time is less than 130 seconds?
A: Using a standard normal distribution table or calculator, we can find the probability that Terri Vogel's lap time is less than 130 seconds.
P(X < 130) = P(Z < 0.122) = 0.5506
This means that there is a 55.06% chance that Terri Vogel's lap time will be less than 130 seconds.
Q: How do I find the interval within which 95% of the data falls?
A: We can use the z-score to find the interval within which 95% of the data falls. For example, if we want to find the interval within which 95% of the data falls, we can use the following formula:
P(129.72 - 2.3 < X < 129.72 + 2.3) = P(-0.58 < Z < 0.58) = 0.7194
This means that there is a 71.94% chance that Terri Vogel's lap time will fall within the interval of 127.42 to 132.02 seconds.
Q: What are some future research directions in this area?
A: Some possible research questions include:
- How does Terri Vogel's lap time distribution change over time?
- What are the factors that affect Terri Vogel's lap time distribution?
- Can we use machine learning algorithms to predict Terri Vogel's lap times?
Q: What are some references for further reading?
A: Some recommended references for further reading include:
- [1] Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
- [2] Johnson, R. A., & Bhattacharyya, G. K. (2014). Statistics: Principles and methods. John Wiley & Sons.
Q: What is the standard deviation of Terri Vogel's lap times?
A: The standard deviation of Terri Vogel's lap times is 2.75 seconds.
Q: How do I calculate the standard deviation?
A: To calculate the standard deviation, you can use the following formula:
σ = √(Σ(xi - μ)^2 / (n - 1))
where xi is each data point, μ is the mean, and n is the number of data points.
For example, if we have the following data points:
| Lap Time | 128.5 | 129.2 | 130.1 | 131.5 | 132.2 | 133.1 | 134.5 |
|---|
We can calculate the standard deviation as follows:
σ = √(Σ(xi - 129.72)^2 / (7 - 1)) = √((128.5 - 129.72)^2 + (129.2 - 129.72)^2 + (130.1 - 129.72)^2 + (131.5 - 129.72)^2 + (132.2 - 129.72)^2 + (133.1 - 129.72)^2 + (134.5 - 129.72)^2) / 6 = √(1.22^2 + 0.52^2 + 0.38^2 + 1.78^2 + 2.48^2 + 3.38^2 + 4.78^2) / 6 = √(1.50 + 0.27 + 0.15 + 3.17 + 6.17 + 11.35 + 22.95) / 6 = √45.46 / 6 = √7.57 = 2.75
This is the standard deviation of Terri Vogel's lap times.