Read First Instructions: The Following Questions

Instructions Read Firstinstructions The Following Questions On The

The following questions are designed to help clarify and solve statistical problems involving the normal distribution, z-scores, and Excel functions. Students are instructed to identify errors in existing solutions, complete partially finished problems, or approach completely blank problems from scratch. The key tasks involve calculating probabilities, percentages, or z-scores using Excel functions such as NORM.S.DIST and NORM.S.INV, and explaining each step in detail for full understanding.

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The assignment focuses on applying the properties of the standard normal distribution to real-world scenarios and computational problems using Excel. Specifically, students are asked to find probabilities that correspond to given z-scores, determine percentages of a population meeting certain criteria, and interpret areas under the normal curve. These tasks involve understanding the relationship between z-scores and cumulative probabilities, as well as correctly setting up and using Excel functions like NORM.S.DIST and NORM.S.INV.

For the first problem, students are asked to compute the probability that a bone density score falls between -1.07 and 2.07, given that the scores follow a standard normal distribution. The student’s partially finished answer involves using Excel to find the cumulative probabilities corresponding to each z-score with the NORM.S.DIST function. To correct and complete this, we need to verify the accuracy of these probabilities, then subtract the smaller from the larger to find the probability that the score is within this interval. The correct approach involves recognizing that the cumulative probability from the left at z = -1.07 is approximately 0.1423, and at z = 2.07, it is approximately 0.9808. Therefore, the probability between these z-scores is 0.9808 - 0.1423, which equals approximately 0.8385, representing 83.85% of the population.

In the second problem, the goal is to find what percentage of women, whose heights are normally distributed with a mean of 63.8 inches and a standard deviation of 2.6 inches, meet the U.S. Airforce height requirements of between 62 inches and 78 inches. We convert these raw scores into z-scores: for 62 inches, z = (62 - 63.8)/2.6 ≈ -0.692, and for 78 inches, z = (78 - 63.8)/2.6 ≈ 5.077. Using NORM.S.DIST in Excel for these z-scores gives us the cumulative probabilities, which we subtract to find the proportion of women meeting the criteria. The resulting percentage is obtained by multiplying by 100, giving a high percentage of women that satisfy the height constraints, indicating that almost all fall within this range due to the large upper bound z-score.

The third problem asks for the cumulative area under the standard normal curve to the left of z = -0.296, which can be directly found with NORM.S.DIST(-0.296, TRUE). The area to the right of this z-score is then 1 minus this cumulative area, providing the probability that a randomly selected subject has a z-score greater than -0.296.

In the fifth problem, a common misconception is addressed. When given the area from the right under the normal curve, students attempt to find the corresponding z-score. The student incorrectly uses NORM.INV(0.571101,0,1) expecting to get the z-score, but this function finds the z-score for a given cumulative probability from the left. Since the area from the right is 0.571101, the corresponding cumulative area from the left is 1 - 0.571101 ≈ 0.4289. Therefore, the correct setup is NORM.S.INV(0.4289), which yields approximately -0.179, the actual z-score for that area.

The sixth problem involves finding the percentage of earthquakes with magnitudes less than 2.000, considering the earthquake magnitudes follow a normal distribution with a mean of 1.184 and a standard deviation of 0.587. The process involves converting 2.000 to a z-score: z = (2.000 - 1.184)/0.587 ≈ 1.317. Using NORM.DIST(1.317, 1.184, 0.587, TRUE) yields the probability of an earthquake being less than 2.000, which can then be expressed as a percentage.

This comprehensive understanding of normal distribution calculations, Excel functions, and probability interpretation forms the core analytical skill set for solving these types of problems effectively, with particular emphasis on correct function setup, probability and z-score transformations, and logical reasoning about measures under the curve.

References

  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (9th ed.). Cengage Learning.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics (8th ed.). W. H. Freeman & Company.
  • Urdan, T. (2016). Statistics in Plain English (4th ed.). Routledge.
  • Everitt, B. S. (2002). The Cambridge Dictionary of Statistics. Cambridge University Press.
  • Microsoft. (2023). NORM.S.DIST function. Microsoft Support. https://support.microsoft.com/en-us/excel
  • Microsoft. (2023). NORM.S.INV function. Microsoft Support. https://support.microsoft.com/en-us/excel
  • CDC. (2022). Normal distribution overview. Centers for Disease Control and Prevention. https://www.cdc.gov
  • OpenStax. (2021). Statistics. Rice University. https://openstax.org/books/statistics
  • Statistics Solutions. (2020). Using Normal Distribution in Research. https://www.statisticssolutions.com/
  • Hogg, R. V., McKean, J., & Craig, A. T. (2013). Introduction to Mathematical Statistics (7th ed.). Pearson.

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