Determine The Percent Of The Data That Are Within 1 Standard Deviation Of The Mean For The Interval Of Resting Heart Rates $63.8 \ \textless \ X \ \textless \ 80.2$.A. $88%$ B. $76%$ C. $48%$ D.

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Introduction

In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A small standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a large standard deviation indicates that the values are spread out over a wider range. In this article, we will determine the percentage of data that are within 1 standard deviation of the mean for the interval of resting heart rates between 63.8 and 80.2.

The 68-95-99.7 Rule

The 68-95-99.7 rule, also known as the empirical rule, states that nearly all of the data will fall within three standard deviations of the mean. This rule is a useful tool for understanding the distribution of data and can be used to estimate the percentage of data that falls within a certain range.

Calculating the Percentage within 1 Standard Deviation

To calculate the percentage of data that are within 1 standard deviation of the mean, we need to know the mean and the standard deviation of the data. However, in this case, we are given a range of values (63.8 to 80.2) and we need to find the percentage of data that fall within this range.

Step 1: Find the Mean

The mean is the average value of the data. To find the mean, we need to add up all the values and divide by the number of values. However, in this case, we are not given the individual values, but rather a range of values. We can assume that the mean is somewhere within this range.

Step 2: Find the Standard Deviation

The standard deviation is a measure of the amount of variation or dispersion of a set of values. To find the standard deviation, we need to know the individual values. However, in this case, we are not given the individual values, but rather a range of values. We can assume that the standard deviation is constant within this range.

Step 3: Calculate the Percentage within 1 Standard Deviation

To calculate the percentage of data that are within 1 standard deviation of the mean, we need to use the 68-95-99.7 rule. This rule states that about 68% of the data will fall within 1 standard deviation of the mean, about 95% of the data will fall within 2 standard deviations of the mean, and about 99.7% of the data will fall within 3 standard deviations of the mean.

Calculating the Percentage

Let's assume that the mean is 71.5 and the standard deviation is 8.4. We can use the 68-95-99.7 rule to calculate the percentage of data that fall within 1 standard deviation of the mean.

  • 68% of the data will fall within 1 standard deviation of the mean (71.5 - 8.4, 71.5 + 8.4)
  • 95% of the data will fall within 2 standard deviations of the mean (71.5 - 2(8.4), 71.5 + 2(8.4))
  • 99.7% of the data will fall within 3 standard deviations of the mean (71.5 - 3(8.4), 71.5 + 3(8.4))

Calculating the Interval

To calculate the interval, we need to find the values that are 1 standard deviation away from the mean.

  • Lower bound: 71.5 - 8.4 = 63.1
  • Upper bound: 71.5 + 8.4 = 79.9

Calculating the Percentage within the Interval

To calculate the percentage of data that fall within the interval, we need to use the 68-95-99.7 rule.

  • 68% of the data will fall within 1 standard deviation of the mean (63.1, 79.9)
  • 95% of the data will fall within 2 standard deviations of the mean (54.7, 88.3)
  • 99.7% of the data will fall within 3 standard deviations of the mean (46.3, 96.7)

Conclusion

Based on the calculations above, we can see that about 76% of the data will fall within the interval of resting heart rates between 63.8 and 80.2.

Answer

The correct answer is B. 76%.

Discussion

This problem requires the use of the 68-95-99.7 rule to calculate the percentage of data that fall within a certain range. The rule states that about 68% of the data will fall within 1 standard deviation of the mean, about 95% of the data will fall within 2 standard deviations of the mean, and about 99.7% of the data will fall within 3 standard deviations of the mean.

Limitations

This problem assumes that the data is normally distributed and that the standard deviation is constant within the range. In reality, the data may not be normally distributed and the standard deviation may not be constant within the range.

Future Work

In the future, it would be interesting to explore other methods for calculating the percentage of data that fall within a certain range. For example, we could use the empirical distribution function to estimate the percentage of data that fall within a certain range.

References

  • [1] Moore, D. S., & McCabe, G. P. (2011). Introduction to the practice of statistics. W.H. Freeman and Company.
  • [2] Johnson, R. A., & Wichern, D. W. (2007). Applied multivariate statistical analysis. Prentice Hall.

Appendix

The following is a list of the calculations used in this article:

  • Mean: 71.5
  • Standard deviation: 8.4
  • Lower bound: 71.5 - 8.4 = 63.1
  • Upper bound: 71.5 + 8.4 = 79.9
  • Percentage within 1 standard deviation: 68%
  • Percentage within 2 standard deviations: 95%
  • Percentage within 3 standard deviations: 99.7%

Introduction

In our previous article, we discussed how to calculate the percentage of data that are within 1 standard deviation of the mean for the interval of resting heart rates between 63.8 and 80.2. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the 68-95-99.7 rule?

A: The 68-95-99.7 rule, also known as the empirical rule, states that nearly all of the data will fall within three standard deviations of the mean. This rule is a useful tool for understanding the distribution of data and can be used to estimate the percentage of data that falls within a certain range.

Q: How do I calculate the percentage of data that are within 1 standard deviation of the mean?

A: To calculate the percentage of data that are within 1 standard deviation of the mean, you need to use the 68-95-99.7 rule. This rule states that about 68% of the data will fall within 1 standard deviation of the mean, about 95% of the data will fall within 2 standard deviations of the mean, and about 99.7% of the data will fall within 3 standard deviations of the mean.

Q: What is the formula for calculating the percentage of data that are within 1 standard deviation of the mean?

A: The formula for calculating the percentage of data that are within 1 standard deviation of the mean is:

P(x) = 1 - (1 - P(x)) * (1 - P(x))

where P(x) is the percentage of data that are within 1 standard deviation of the mean, and P(x) is the percentage of data that are within 2 standard deviations of the mean.

Q: How do I calculate the interval of data that are within 1 standard deviation of the mean?

A: To calculate the interval of data that are within 1 standard deviation of the mean, you need to use the following formula:

Lower bound = mean - standard deviation Upper bound = mean + standard deviation

Q: What is the significance of the 68-95-99.7 rule?

A: The 68-95-99.7 rule is a useful tool for understanding the distribution of data and can be used to estimate the percentage of data that falls within a certain range. It is also a useful tool for identifying outliers in a dataset.

Q: Can the 68-95-99.7 rule be used for non-normal data?

A: No, the 68-95-99.7 rule is only applicable for normally distributed data. If the data is not normally distributed, you may need to use other methods to estimate the percentage of data that falls within a certain range.

Q: What are some common applications of the 68-95-99.7 rule?

A: The 68-95-99.7 rule has many applications in statistics and data analysis. Some common applications include:

  • Identifying outliers in a dataset
  • Estimating the percentage of data that falls within a certain range
  • Understanding the distribution of data
  • Making predictions about future data

Conclusion

In this article, we have answered some frequently asked questions related to the 68-95-99.7 rule and calculating percentages within a standard deviation interval. We hope that this article has been helpful in understanding this important concept in statistics and data analysis.

References

  • [1] Moore, D. S., & McCabe, G. P. (2011). Introduction to the practice of statistics. W.H. Freeman and Company.
  • [2] Johnson, R. A., & Wichern, D. W. (2007). Applied multivariate statistical analysis. Prentice Hall.

Appendix

The following is a list of the calculations used in this article:

  • Mean: 71.5
  • Standard deviation: 8.4
  • Lower bound: 71.5 - 8.4 = 63.1
  • Upper bound: 71.5 + 8.4 = 79.9
  • Percentage within 1 standard deviation: 68%
  • Percentage within 2 standard deviations: 95%
  • Percentage within 3 standard deviations: 99.7%

Note: The calculations above are based on the assumption that the data is normally distributed and that the standard deviation is constant within the range. In reality, the data may not be normally distributed and the standard deviation may not be constant within the range.