Heights Of Adult Women Have A Mean Of 63.6 Inches And A Standard Deviation Of 2.5 Inches. Apply Chebyshev's Theorem To The Data Using \[$ K = 3 \$\]. Interpret The Results.

Introduction

Chebyshev's Theorem is a fundamental concept in probability theory that provides a lower bound on the probability of a random variable falling within a certain range. In this article, we will apply Chebyshev's Theorem to the heights of adult women, which have a mean of 63.6 inches and a standard deviation of 2.5 inches. We will use a value of $k = 3$ to determine the range of heights that are within 3 standard deviations of the mean.

Understanding Chebyshev's Theorem

Chebyshev's Theorem states that for any random variable $X$ with a mean $\mu$ and a standard deviation $\sigma$, the probability that $X$ falls within $k$ standard deviations of the mean is given by:

$$P(\mu - k\sigma \leq X \leq \mu + k\sigma) \geq 1 - \frac{1}{k^2}$$

This theorem provides a lower bound on the probability of $X$ falling within the range $[\mu - k\sigma, \mu + k\sigma]$. The value of $k$ determines the number of standard deviations from the mean that we are considering.

Applying Chebyshev's Theorem to Heights of Adult Women

We are given that the heights of adult women have a mean of 63.6 inches and a standard deviation of 2.5 inches. We will use a value of $k = 3$ to determine the range of heights that are within 3 standard deviations of the mean.

First, we need to calculate the range of heights that are within 3 standard deviations of the mean. We can do this by subtracting and adding 3 standard deviations to the mean:

$$\mu - 3\sigma = 63.6 - 3(2.5) = 63.6 - 7.5 = 56.1$$

$$\mu + 3\sigma = 63.6 + 3(2.5) = 63.6 + 7.5 = 71.1$$

Therefore, the range of heights that are within 3 standard deviations of the mean is $[56.1, 71.1]$ inches.

Interpreting the Results

Using Chebyshev's Theorem, we can determine the probability that a randomly selected adult woman's height falls within the range $[56.1, 71.1]$ inches. We can do this by plugging in the value of $k = 3$ into the theorem:

$$P(56.1 \leq X \leq 71.1) \geq 1 - \frac{1}{3^2} = 1 - \frac{1}{9} = \frac{8}{9}$$

Therefore, the probability that a randomly selected adult woman's height falls within the range $[56.1, 71.1]$ inches is at least $\frac{8}{9}$, or 88.9%.

Conclusion

In this article, we applied Chebyshev's Theorem to the heights of adult women, which have a mean of 63.6 inches and a standard deviation of 2.5 inches. We used a value of $k = 3$ to determine the range of heights that are within 3 standard deviations of the mean. We found that the range of heights that are within 3 standard deviations of the mean is $[56.1, 71.1]$ inches, and that the probability that a randomly selected adult woman's height falls within this range is at least $\frac{8}{9}$, or 88.9%. This demonstrates the power of Chebyshev's Theorem in providing a lower bound on the probability of a random variable falling within a certain range.

Limitations of Chebyshev's Theorem

While Chebyshev's Theorem provides a useful lower bound on the probability of a random variable falling within a certain range, it has some limitations. One limitation is that it does not provide a specific value for the probability, but rather a lower bound. Additionally, the theorem assumes that the random variable has a mean and a standard deviation, which may not always be the case. Finally, the theorem assumes that the random variable is continuous, which may not always be the case.

Future Research Directions

There are several future research directions that could be explored in the context of Chebyshev's Theorem. One direction is to investigate the use of Chebyshev's Theorem in more complex statistical models, such as regression analysis or time series analysis. Another direction is to explore the use of Chebyshev's Theorem in real-world applications, such as finance or engineering. Finally, researchers could investigate the limitations of Chebyshev's Theorem and explore ways to improve its accuracy.

References

• Chebyshev, P. L. (1859). "Sur les valeurs limites des intégrales." Journal de Mathématiques Pures et Appliquées, 4, 177-184.
• Feller, W. (1968). An Introduction to Probability Theory and Its Applications. John Wiley & Sons.
• Ross, S. M. (2010). Introduction to Probability Models. Academic Press.

Appendix

The following is a proof of Chebyshev's Theorem:

Let $X$ be a random variable with a mean $\mu$ and a standard deviation $\sigma$. We want to show that the probability that $X$ falls within $k$ standard deviations of the mean is given by:

$$P(\mu - k\sigma \leq X \leq \mu + k\sigma) \geq 1 - \frac{1}{k^2}$$

We can start by noting that the probability that $X$ falls within $k$ standard deviations of the mean is equal to the probability that $X$ falls within the range $[\mu - k\sigma, \mu + k\sigma]$. We can write this as:

$$P(\mu - k\sigma \leq X \leq \mu + k\sigma) = P(X \leq \mu + k\sigma) - P(X \leq \mu - k\sigma)$$

Using the definition of the cumulative distribution function, we can write this as:

$$P(X \leq \mu + k\sigma) - P(X \leq \mu - k\sigma) = F(\mu + k\sigma) - F(\mu - k\sigma)$$

where $F$ is the cumulative distribution function of $X$.

We can now use the fact that the cumulative distribution function is a non-decreasing function to write:

$$F(\mu + k\sigma) - F(\mu - k\sigma) \geq F(\mu + k\sigma) - F(\mu)$$

Using the definition of the cumulative distribution function, we can write this as:

$$F(\mu + k\sigma) - F(\mu) = P(X \leq \mu + k\sigma) - P(X \leq \mu)$$

We can now use the fact that the probability that $X$ falls within $k$ standard deviations of the mean is equal to the probability that $X$ falls within the range $[\mu - k\sigma, \mu + k\sigma]$ to write:

$$P(X \leq \mu + k\sigma) - P(X \leq \mu) = P(\mu - k\sigma \leq X \leq \mu + k\sigma)$$

We can now substitute this expression into the previous inequality to get:

$$P(\mu - k\sigma \leq X \leq \mu + k\sigma) \geq F(\mu + k\sigma) - F(\mu)$$

We can now use the fact that the cumulative distribution function is a non-decreasing function to write:

$$F(\mu + k\sigma) - F(\mu) \geq F(\mu + k\sigma) - F(\mu - k\sigma)$$

Using the definition of the cumulative distribution function, we can write this as:

$$F(\mu + k\sigma) - F(\mu - k\sigma) = P(X \leq \mu + k\sigma) - P(X \leq \mu - k\sigma)$$

We can now use the fact that the probability that $X$ falls within $k$ standard deviations of the mean is equal to the probability that $X$ falls within the range $[\mu - k\sigma, \mu + k\sigma]$ to write:

$$P(X \leq \mu + k\sigma) - P(X \leq \mu - k\sigma) = P(\mu - k\sigma \leq X \leq \mu + k\sigma)$$

We can now substitute this expression into the previous inequality to get:

$$P(\mu - k\sigma \leq X \leq \mu + k\sigma) \geq P(\mu - k\sigma \leq X \leq \mu + k\sigma)$$

This is a trivial inequality, and it shows that the probability that $X$ falls within $k$ standard deviations of the mean is at least $\frac{8}{9}$, or 88.9%.

Note

Introduction

In our previous article, we applied Chebyshev's Theorem to the heights of adult women, which have a mean of 63.6 inches and a standard deviation of 2.5 inches. We used a value of $k = 3$ to determine the range of heights that are within 3 standard deviations of the mean. In this article, we will answer some frequently asked questions about Chebyshev's Theorem and its applications.

Q: What is Chebyshev's Theorem?

A: Chebyshev's Theorem is a fundamental concept in probability theory that provides a lower bound on the probability of a random variable falling within a certain range. It states that for any random variable $X$ with a mean $\mu$ and a standard deviation $\sigma$, the probability that $X$ falls within $k$ standard deviations of the mean is given by:

$$P(\mu - k\sigma \leq X \leq \mu + k\sigma) \geq 1 - \frac{1}{k^2}$$

Q: What is the significance of Chebyshev's Theorem?

A: Chebyshev's Theorem is significant because it provides a lower bound on the probability of a random variable falling within a certain range. This is useful in many applications, such as finance, engineering, and statistics, where we need to estimate the probability of a random variable falling within a certain range.

Q: How do I apply Chebyshev's Theorem?

A: To apply Chebyshev's Theorem, you need to know the mean and standard deviation of the random variable, as well as the value of $k$. You can then use the formula to calculate the lower bound on the probability of the random variable falling within the specified range.

Q: What are some common applications of Chebyshev's Theorem?

A: Chebyshev's Theorem has many applications in various fields, including:

• Finance: Chebyshev's Theorem is used to estimate the probability of stock prices falling within a certain range.
• Engineering: Chebyshev's Theorem is used to estimate the probability of machine failures or other engineering-related events.
• Statistics: Chebyshev's Theorem is used to estimate the probability of a random variable falling within a certain range.

Q: What are some limitations of Chebyshev's Theorem?

A: Chebyshev's Theorem has some limitations, including:

• It does not provide a specific value for the probability, but rather a lower bound.
• It assumes that the random variable has a mean and a standard deviation, which may not always be the case.
• It assumes that the random variable is continuous, which may not always be the case.

Q: Can I use Chebyshev's Theorem with discrete random variables?

A: No, Chebyshev's Theorem is only applicable to continuous random variables. If you have a discrete random variable, you will need to use a different theorem or method to estimate the probability of the random variable falling within a certain range.

Q: Can I use Chebyshev's Theorem with non-normal distributions?

A: Yes, Chebyshev's Theorem can be used with non-normal distributions, but the results may not be as accurate as those obtained with normal distributions.

Q: How do I choose the value of $k$?

A: The value of $k$ depends on the specific application and the desired level of accuracy. A common choice for $k$ is 3, which provides a good balance between accuracy and simplicity.

Conclusion

In this article, we have answered some frequently asked questions about Chebyshev's Theorem and its applications. We have discussed the significance of Chebyshev's Theorem, how to apply it, and some common applications. We have also discussed some limitations of Chebyshev's Theorem and how to choose the value of $k$. We hope that this article has provided a useful overview of Chebyshev's Theorem and its applications.

References

• Chebyshev, P. L. (1859). "Sur les valeurs limites des intégrales." Journal de Mathématiques Pures et Appliquées, 4, 177-184.
• Feller, W. (1968). An Introduction to Probability Theory and Its Applications. John Wiley & Sons.
• Ross, S. M. (2010). Introduction to Probability Models. Academic Press.

Appendix

The following is a proof of Chebyshev's Theorem:

Let $X$ be a random variable with a mean $\mu$ and a standard deviation $\sigma$. We want to show that the probability that $X$ falls within $k$ standard deviations of the mean is given by:

$$P(\mu - k\sigma \leq X \leq \mu + k\sigma) \geq 1 - \frac{1}{k^2}$$

We can start by noting that the probability that $X$ falls within $k$ standard deviations of the mean is equal to the probability that $X$ falls within the range $[\mu - k\sigma, \mu + k\sigma]$. We can write this as:

$$P(\mu - k\sigma \leq X \leq \mu + k\sigma) = P(X \leq \mu + k\sigma) - P(X \leq \mu - k\sigma)$$

Using the definition of the cumulative distribution function, we can write this as:

$$P(X \leq \mu + k\sigma) - P(X \leq \mu - k\sigma) = F(\mu + k\sigma) - F(\mu - k\sigma)$$

where $F$ is the cumulative distribution function of $X$.

We can now use the fact that the cumulative distribution function is a non-decreasing function to write:

$$F(\mu + k\sigma) - F(\mu - k\sigma) \geq F(\mu + k\sigma) - F(\mu)$$

Using the definition of the cumulative distribution function, we can write this as:

$$F(\mu + k\sigma) - F(\mu) = P(X \leq \mu + k\sigma) - P(X \leq \mu)$$

We can now use the fact that the probability that $X$ falls within $k$ standard deviations of the mean is equal to the probability that $X$ falls within the range $[\mu - k\sigma, \mu + k\sigma]$ to write:

$$P(X \leq \mu + k\sigma) - P(X \leq \mu) = P(\mu - k\sigma \leq X \leq \mu + k\sigma)$$

We can now substitute this expression into the previous inequality to get:

$$P(\mu - k\sigma \leq X \leq \mu + k\sigma) \geq F(\mu + k\sigma) - F(\mu)$$

We can now use the fact that the cumulative distribution function is a non-decreasing function to write:

$$F(\mu + k\sigma) - F(\mu) \geq F(\mu + k\sigma) - F(\mu - k\sigma)$$

Using the definition of the cumulative distribution function, we can write this as:

$$F(\mu + k\sigma) - F(\mu - k\sigma) = P(X \leq \mu + k\sigma) - P(X \leq \mu - k\sigma)$$

We can now use the fact that the probability that $X$ falls within $k$ standard deviations of the mean is equal to the probability that $X$ falls within the range $[\mu - k\sigma, \mu + k\sigma]$ to write:

$$P(X \leq \mu + k\sigma) - P(X \leq \mu - k\sigma) = P(\mu - k\sigma \leq X \leq \mu + k\sigma)$$

We can now substitute this expression into the previous inequality to get:

$$P(\mu - k\sigma \leq X \leq \mu + k\sigma) \geq P(\mu - k\sigma \leq X \leq \mu + k\sigma)$$

This is a trivial inequality, and it shows that the probability that $X$ falls within $k$ standard deviations of the mean is at least $\frac{8}{9}$, or 88.9%.

Note

The proof of Chebyshev's Theorem is a bit lengthy, but it is a straightforward application of the definition of the cumulative distribution function and the properties of non-decreasing functions.