Math Problem Statement

A forester measured 3535 of the trees in a large woods that is up for sale. He found a mean diameter of 10.710.7 inches and a standard deviation of 4.74.7 inches. Suppose that these trees provide an accurate description of the whole forest and that a Normal model applies. Question content area bottom Part 1 ​a) Choose the correct Normal model for tree diameters. A. font size decreased by 3 10.710.7 font size decreased by 2 mu plus sigmaμ+σ font size decreased by 2 mu plus 2 sigmaμ+2σ font size decreased by 2 mu plus 3 sigmaμ+3σ font size decreased by 2 mu minus sigmaμ−σ font size decreased by 2 mu minus 2 sigmaμ−2σ font size decreased by 2 mu minus 3 sigmaμ−3σ font size decreased by 2 muμ font size decreased by 3 15.415.4 font size decreased by 3 6.06.0 font size decreased by 3 20.120.1 font size decreased by 3 1.31.3 font size decreased by 3 negative 3.4−3.4 font size decreased by 3 24.824.8 68% 95% 99.7%

A bell-shaped curve with horizontal axis from less than negative 3.4 to 24.8 plus in intervals of 4.7 begins just above the horizontal axis, increases at an increasing and then decreasing rate to its maximum at 10.7, labeled "mu," and decreases at an increasing and then decreasing rate approaching the horizontal axis. The curve is symmetric. The area below the curve is subdivided into regions by the intervals as follows: the region from mu minus sigma, 6.0, to mu plus sigma, 15.4, is labeled "99.7%"; the region from mu minus 2 times sigma, 1.3, to mu plus 2 times sigma, 20.1, is labeled "95%"; the region from mu minus 3 times sigma, negative 3.4, to mu plus 3 times sigma, 24.8, is labeled "68%." B. font size decreased by 3 10.710.7 font size decreased by 2 mu plus sigmaμ+σ font size decreased by 2 mu plus 2 sigmaμ+2σ font size decreased by 2 mu plus 3 sigmaμ+3σ font size decreased by 2 mu minus sigmaμ−σ font size decreased by 2 mu minus 2 sigmaμ−2σ font size decreased by 2 mu minus 3 sigmaμ−3σ font size decreased by 2 muμ font size decreased by 3 15.415.4 font size decreased by 3 6.06.0 font size decreased by 3 20.120.1 font size decreased by 3 1.31.3 font size decreased by 3 negative 3.4−3.4 font size decreased by 3 24.824.8 68% 95% 99.7%

Solution

To determine the correct Normal model for tree diameters based on the provided data, we need to analyze the mean diameter and the standard deviation.

  1. Given Values:

    • Mean (μ) = 10.7 inches
    • Standard Deviation (σ) = 4.7 inches

  2. Calculate the Key Values:

    • μ - 3σ: 10.7−3(4.7)=10.7−14.1=−3.410.7 - 3(4.7) = 10.7 - 14.1 = -3.4
    • μ - 2σ: 10.7−2(4.7)=10.7−9.4=1.310.7 - 2(4.7) = 10.7 - 9.4 = 1.3
    • μ - σ: 10.7−4.7=6.010.7 - 4.7 = 6.0
    • μ + σ: 10.7+4.7=15.410.7 + 4.7 = 15.4
    • μ + 2σ: 10.7+9.4=20.110.7 + 9.4 = 20.1
    • μ + 3σ: 10.7+14.1=24.810.7 + 14.1 = 24.8

  3. Model Distribution:

    • The Normal model is symmetric around the mean (10.7 inches).
    • The percentages associated with the intervals are typically:
      • 68% of data within μ ± σ
      • 95% of data within μ ± 2σ
      • 99.7% of data within μ ± 3σ

  4. Selecting the Model:

    • Interval for 68%: From 6.0 to 15.4 (μ ± σ)
    • Interval for 95%: From 1.3 to 20.1 (μ ± 2σ)
    • Interval for 99.7%: From -3.4 to 24.8 (μ ± 3σ)

Conclusion

The correct Normal model is represented by the specified intervals with percentages indicated. Thus, based on the data, the choice corresponds to the values outlined in option A.

If you have any questions or need further details, feel free to ask!

Follow-up Questions

  1. How does the Normal distribution apply to real-world data?
  2. What is the significance of the 68-95-99.7 rule in statistics?
  3. How would you interpret a tree diameter of 15 inches in this model?
  4. What factors could affect the accuracy of this Normal model?
  5. How would you visualize this Normal distribution graphically?

Tip: Always visualize data using histograms or normal probability plots to assess normality before applying statistical models.

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Math Problem Analysis

Mathematical Concepts

Statistics
Normal Distribution
Descriptive Statistics

Formulas

μ - 3σ
μ - 2σ
μ - σ
μ + σ
μ + 2σ
μ + 3σ

Theorems

68-95-99.7 Rule

Suitable Grade Level

Grades 9-12

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