The Mean IQ Score Of Adults Is 100, With A Standard Deviation Of 15. Use The Empirical Rule To Find The Percentage Of Adults With Scores Between 70 And 130. Assume The Data Set Has A Bell-shaped Distribution.A) 99.7% B) 100% C) 95% D) 68%

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The Empirical Rule: Understanding the Distribution of IQ Scores

Introduction

The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical concept that describes the distribution of data in a normal distribution. It 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. In this article, we will apply the Empirical Rule to find the percentage of adults with IQ scores between 70 and 130, given a mean IQ score of 100 and a standard deviation of 15.

Understanding the Empirical Rule

The Empirical Rule is a fundamental concept in statistics that helps us understand the distribution of data in a normal distribution. It is based on the idea that most of the data points in a normal distribution are clustered around the mean, with fewer data points falling further away from the mean. The rule is as follows:

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

Applying the Empirical Rule to IQ Scores

To apply the Empirical Rule to IQ scores, we need to understand the distribution of IQ scores. The mean IQ score is 100, and the standard deviation is 15. We are interested in finding the percentage of adults with IQ scores between 70 and 130.

First, let's calculate the number of standard deviations between 70 and 130.

  • The mean IQ score is 100.
  • The standard deviation is 15.
  • The IQ score of 70 is 2 standard deviations below the mean (100 - 2 x 15 = 70).
  • The IQ score of 130 is 2 standard deviations above the mean (100 + 2 x 15 = 130).

Since the IQ score of 70 is 2 standard deviations below the mean, and the IQ score of 130 is 2 standard deviations above the mean, we can conclude that the IQ scores between 70 and 130 fall within two standard deviations of the mean.

Using the Empirical Rule to Find the Percentage

Now that we have established that the IQ scores between 70 and 130 fall within two standard deviations of the mean, we can use the Empirical Rule to find the percentage of adults with IQ scores in this range.

According to the Empirical Rule, about 95% of the data falls within two standard deviations of the mean. Therefore, we can conclude that about 95% of adults have IQ scores between 70 and 130.

Conclusion

In conclusion, the Empirical Rule is a powerful tool for understanding the distribution of data in a normal distribution. By applying the Empirical Rule to IQ scores, we can conclude that about 95% of adults have IQ scores between 70 and 130. This is a significant finding, as it highlights the importance of understanding the distribution of data in a normal distribution.

Answer

The correct answer is C) 95%.

Discussion

The Empirical Rule is a fundamental concept in statistics that helps us understand the distribution of data in a normal distribution. By applying the Empirical Rule to IQ scores, we can conclude that about 95% of adults have IQ scores between 70 and 130. This is a significant finding, as it highlights the importance of understanding the distribution of data in a normal distribution.

Additional Information

  • The Empirical Rule is also known as the 68-95-99.7 rule.
  • The Empirical Rule is based on the idea that most of the data points in a normal distribution are clustered around the mean, with fewer data points falling further away from the mean.
  • The Empirical Rule is a useful tool for understanding the distribution of data in a normal distribution.

References

  • Moore, D. S., & McCabe, G. P. (2011). Introduction to the practice of statistics. W.H. Freeman and Company.
  • Larson, R. E., & Farber, B. E. (2018). Elementary statistics: Picturing the world. Cengage Learning.

Key Terms

  • Empirical Rule: A statistical concept that describes the distribution of data in a normal distribution.
  • Normal distribution: A type of distribution that is symmetric around the mean.
  • Standard deviation: A measure of the amount of variation or dispersion of a set of values.
  • Mean: The average value of a set of values.
    The Empirical Rule: A Q&A Guide

Introduction

The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical concept that describes the distribution of data in a normal distribution. It is a fundamental concept in statistics that helps us understand the distribution of data in a normal distribution. In this article, we will answer some frequently asked questions about the Empirical Rule.

Q: What is the Empirical Rule?

A: The Empirical Rule is a statistical concept that describes the distribution of data in a normal distribution. It 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.

Q: What is a normal distribution?

A: A normal distribution is a type of distribution that is symmetric around the mean. It is a bell-shaped distribution where the majority of the data points are clustered around the mean, with fewer data points falling further away from the mean.

Q: What is a standard deviation?

A: A standard deviation is a measure of the amount of variation or dispersion of a set of values. It is a way to measure how spread out the data is from the mean.

Q: How do I apply the Empirical Rule to my data?

A: To apply the Empirical Rule to your data, you need to calculate the mean and standard deviation of your data. Then, you can use the Empirical Rule to determine the percentage of data that falls within one, two, or three standard deviations of the mean.

Q: What is the significance of the Empirical Rule?

A: The Empirical Rule is a significant concept in statistics because it helps us understand the distribution of data in a normal distribution. It is a useful tool for understanding how data is distributed and for making predictions about future data.

Q: Can I use the Empirical Rule with non-normal data?

A: No, the Empirical Rule is only applicable to normal data. If your data is not normally distributed, you may need to use other statistical methods to understand the distribution of your data.

Q: How do I know if my data is normally distributed?

A: You can use a normality test to determine if your data is normally distributed. Some common normality tests include the Shapiro-Wilk test and the Kolmogorov-Smirnov test.

Q: What are some common applications of the Empirical Rule?

A: The Empirical Rule has many common applications in statistics, including:

  • Understanding the distribution of IQ scores
  • Understanding the distribution of exam scores
  • Understanding the distribution of stock prices
  • Understanding the distribution of weather patterns

Q: Can I use the Empirical Rule to make predictions about future data?

A: Yes, the Empirical Rule can be used to make predictions about future data. By understanding the distribution of your data, you can make predictions about how future data will be distributed.

Q: What are some common misconceptions about the Empirical Rule?

A: Some common misconceptions about the Empirical Rule include:

  • The Empirical Rule only applies to data that is normally distributed.
  • The Empirical Rule is only applicable to large datasets.
  • The Empirical Rule is only used in academic research.

Conclusion

The Empirical Rule is a fundamental concept in statistics that helps us understand the distribution of data in a normal distribution. By answering some frequently asked questions about the Empirical Rule, we can gain a better understanding of this important statistical concept.

Additional Information

  • The Empirical Rule is also known as the 68-95-99.7 rule.
  • The Empirical Rule is based on the idea that most of the data points in a normal distribution are clustered around the mean, with fewer data points falling further away from the mean.
  • The Empirical Rule is a useful tool for understanding the distribution of data in a normal distribution.

References

  • Moore, D. S., & McCabe, G. P. (2011). Introduction to the practice of statistics. W.H. Freeman and Company.
  • Larson, R. E., & Farber, B. E. (2018). Elementary statistics: Picturing the world. Cengage Learning.

Key Terms

  • Empirical Rule: A statistical concept that describes the distribution of data in a normal distribution.
  • Normal distribution: A type of distribution that is symmetric around the mean.
  • Standard deviation: A measure of the amount of variation or dispersion of a set of values.
  • Mean: The average value of a set of values.