The Lengths Of A Lawn Mower Part Are Approximately Normally Distributed With A Given Mean Μ = 4 In. \mu = 4 \text{ In.} Μ = 4 In. And Standard Deviation Σ = 0.2 In. \sigma = 0.2 \text{ In.} Σ = 0.2 In. What Percentage Of The Parts Will Have Lengths Between 3.8 In. And 4.2

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Understanding the Problem

The problem involves finding the percentage of lawn mower parts that will have lengths between 3.8 in. and 4.2 in. given that the lengths are approximately normally distributed with a mean of 4 in. and a standard deviation of 0.2 in. This is a classic problem in statistics that can be solved using the properties of the normal distribution.

Properties of the Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about the mean. It is characterized by two parameters: the mean ($\mu$) and the standard deviation ($\sigma$). The normal distribution is often denoted as $N(\mu, \sigma^2)$.

One of the key properties of the normal distribution is that about 68% of the data points fall within one standard deviation of the mean, about 95% fall within two standard deviations, and about 99.7% fall within three standard deviations.

Standardizing the Data

To solve this problem, we need to standardize the data by converting the given lengths to z-scores. The z-score is a measure of how many standard deviations an observation is away from the mean. It is calculated using the formula:

$$z = \frac{X - \mu}{\sigma}$$

where $X$ is the value of the observation, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Calculating the Z-Scores

We need to calculate the z-scores for the given lengths of 3.8 in. and 4.2 in.

For $X = 3.8$ in.:

$$z = \frac{3.8 - 4}{0.2} = -1$$

For $X = 4.2$ in.:

$$z = \frac{4.2 - 4}{0.2} = 1$$

Using a Standard Normal Distribution Table

We can use a standard normal distribution table (also known as a z-table) to find the probabilities corresponding to the z-scores. The z-table shows the probability that a standard normal random variable $Z$ will take on a value less than or equal to a given z-score.

Finding the Probabilities

Using the z-table, we find that:

• $P(Z \leq -1) = 0.1587$
• $P(Z \leq 1) = 0.8413$

Finding the Percentage of Parts

We want to find the percentage of parts that will have lengths between 3.8 in. and 4.2 in. This is equivalent to finding the probability that a standard normal random variable $Z$ will take on a value between -1 and 1.

We can use the fact that the normal distribution is symmetric about the mean to find this probability. Since $P(Z \leq 1) = 0.8413$, we know that $P(-1 \leq Z \leq 1) = 2 \times (0.8413 - 0.5) = 0.6827$.

Conclusion

Therefore, approximately 68.27% of the parts will have lengths between 3.8 in. and 4.2 in.

Interpretation

This result makes sense in the context of the problem. Since the lengths are approximately normally distributed with a mean of 4 in. and a standard deviation of 0.2 in., we would expect about 68% of the parts to fall within one standard deviation of the mean. This is consistent with the result we obtained.

Limitations

One limitation of this result is that it assumes that the lengths are normally distributed. In reality, the lengths may not be normally distributed, and the result may not be accurate. Additionally, the result assumes that the standard deviation is known, which may not always be the case.

Future Work

Future work could involve investigating the distribution of the lengths and determining whether they are normally distributed. Additionally, future work could involve estimating the standard deviation of the lengths and using this estimate to calculate the percentage of parts that will have lengths between 3.8 in. and 4.2 in.

References

• [1] Moore, D. S., & McCabe, G. P. (2013). Introduction to the practice of statistics. W.H. Freeman and Company.
• [2] Ross, S. M. (2012). Introduction to probability models. Academic Press.

Appendix

The following is a list of the z-scores and their corresponding probabilities:

z-score	probability
-1	0.1587
1	0.8413

Note: The probabilities are approximate and are based on a standard normal distribution table.

Q: What is the normal distribution?

A: The normal distribution is a continuous probability distribution that is symmetric about the mean. It is characterized by two parameters: the mean ($\mu$) and the standard deviation ($\sigma$). The normal distribution is often denoted as $N(\mu, \sigma^2)$.

Q: What are the key properties of the normal distribution?

A: The normal distribution has several key properties, including:

• About 68% of the data points fall within one standard deviation of the mean.
• About 95% of the data points fall within two standard deviations of the mean.
• About 99.7% of the data points fall within three standard deviations of the mean.

Q: How do I calculate the z-score?

A: The z-score is a measure of how many standard deviations an observation is away from the mean. It is calculated using the formula:

$$z = \frac{X - \mu}{\sigma}$$

where $X$ is the value of the observation, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Q: What is the difference between a z-score and a standard deviation?

A: A z-score is a measure of how many standard deviations an observation is away from the mean, while a standard deviation is a measure of the spread of the data.

Q: How do I use a standard normal distribution table (z-table)?

A: A standard normal distribution table (z-table) shows the probability that a standard normal random variable $Z$ will take on a value less than or equal to a given z-score. To use a z-table, you need to find the z-score corresponding to the probability you are interested in.

Q: What is the relationship between the normal distribution and the 68-95-99.7 rule?

A: The 68-95-99.7 rule is a statement of the key properties of the normal distribution. It states that about 68% of the data points fall within one standard deviation of the mean, about 95% of the data points fall within two standard deviations of the mean, and about 99.7% of the data points fall within three standard deviations of the mean.

Q: Can the normal distribution be used to model real-world data?

A: Yes, the normal distribution can be used to model real-world data. However, it is essential to check whether the data is normally distributed before using the normal distribution to model it.

Q: What are some common applications of the normal distribution?

A: The normal distribution has many common applications, including:

• Modeling the distribution of exam scores
• Modeling the distribution of heights and weights
• Modeling the distribution of stock prices
• Modeling the distribution of errors in measurement

Q: What are some common misconceptions about the normal distribution?

A: Some common misconceptions about the normal distribution include:

• The normal distribution is always symmetric.
• The normal distribution is always bell-shaped.
• The normal distribution is always continuous.

Q: How do I determine whether a dataset is normally distributed?

A: To determine whether a dataset is normally distributed, you can use statistical tests such as the Shapiro-Wilk test or the Kolmogorov-Smirnov test. You can also use graphical methods such as Q-Q plots to visualize the distribution of the data.

Q: What are some common statistical tests used to determine whether a dataset is normally distributed?

A: Some common statistical tests used to determine whether a dataset is normally distributed include:

• Shapiro-Wilk test
• Kolmogorov-Smirnov test
• Q-Q plots

Q: What are some common applications of the normal distribution in real-world scenarios?

A: The normal distribution has many common applications in real-world scenarios, including:

• Modeling the distribution of exam scores
• Modeling the distribution of heights and weights
• Modeling the distribution of stock prices
• Modeling the distribution of errors in measurement

Q: What are some common challenges associated with using the normal distribution?

A: Some common challenges associated with using the normal distribution include:

• The normal distribution may not be a good fit for the data.
• The normal distribution may not be symmetric.
• The normal distribution may not be continuous.

Q: How do I choose the correct statistical test for determining whether a dataset is normally distributed?

A: To choose the correct statistical test for determining whether a dataset is normally distributed, you need to consider the following factors:

• The size of the dataset
• The distribution of the data
• The type of data

Q: What are some common statistical software packages used to analyze data that follows a normal distribution?

A: Some common statistical software packages used to analyze data that follows a normal distribution include:

• R
• Python
• SAS
• SPSS

Q: What are some common applications of the normal distribution in finance?

A: The normal distribution has many common applications in finance, including:

• Modeling the distribution of stock prices
• Modeling the distribution of returns
• Modeling the distribution of risks

Q: What are some common challenges associated with using the normal distribution in finance?

A: Some common challenges associated with using the normal distribution in finance include:

• The normal distribution may not be a good fit for the data.
• The normal distribution may not be symmetric.
• The normal distribution may not be continuous.

Q: How do I choose the correct statistical test for determining whether a dataset is normally distributed in finance?

A: To choose the correct statistical test for determining whether a dataset is normally distributed in finance, you need to consider the following factors:

• The size of the dataset
• The distribution of the data
• The type of data

Q: What are some common statistical software packages used to analyze data that follows a normal distribution in finance?

A: Some common statistical software packages used to analyze data that follows a normal distribution in finance include:

• R
• Python
• SAS
• SPSS