Deliverable 02 Worksheet Instructions: The Following Workshe

Deliverable 02 Worksheetinstructions: The Following Worksheet Is Show

The worksheet presents multiple problems related to normal distribution, z-scores, and percentages, along with some unrelated text about Walmart's strategy and acquisitions. Your task is to help interpret and solve each of these statistical problems in detail, explaining all steps and calculations clearly, and providing corrections where needed. Focus on addressing each problem, completing incomplete solutions, correcting errors, and illustrating the reasoning behind each step. Do not include the unrelated Walmart business text in your response. Your goal is to produce a comprehensive, well-structured, academic report addressing the statistical questions using appropriate methods such as standard normal tables, Excel functions, and probability concepts, supported by credible references. Ensure your explanations are detailed enough for a student unfamiliar with the concepts to understand your reasoning and the procedures involved.

Paper For Above instruction

Understanding the application of the normal distribution, z-scores, and probability calculations is fundamental in statistics, as these concepts underpin many real-world data analysis scenarios. In this paper, we explore several problems involving the standard normal distribution, highlighting the methods to calculate probabilities, interpret z-scores, and understand percentile ranks. Each problem is discussed in detail, emphasizing the steps involved, common pitfalls, and interpretations critical for accurate statistical analysis.

Problem 1: Find the probability that a bone density test score lies between -1.63 and 1.96, given that the scores follow a standard normal distribution. The student’s initial approach involves using Excel to find cumulative probabilities corresponding to these z-scores. For -1.63, the probability from the left (area under the curve) is approximately 0.0516, and for 1.96, it is approximately 0.9750. To complete this problem, we subtract the cumulative probability at -1.63 from that at 1.96:

Probability = P(-1.63

This indicates that about 92.34% of subjects have bone density scores falling within this range. It demonstrates the use of the standard normal table or a calculator to find cumulative probabilities and the importance of understanding symmetry in the normal distribution (since Φ(-z) = 1 - Φ(z)).

Problem 2: The U.S. Airforce has height requirements for pilots between 64 inches and 77 inches. Women’s heights are normally distributed with a mean of 65.2 inches and a standard deviation of 3.53 inches. The task is to find the percentage of women who meet the height requirement. The approach involves calculating z-scores for the lower and upper bounds:

For the lower bound (64 inches):

z = (64 - 65.2) / 3.53 ≈ -0.34

For the upper bound (77 inches):

z = (77 - 65.2) / 3.53 ≈ 3.45

Using a standard normal table or calculator, the cumulative probability for z = -0.34 is approximately 0.366, and for z = 3.45, it is about 0.9987. Therefore, the percentage of women within this height range is:

Percentage = (0.9987 - 0.366) × 100% ≈ 63.21%

This means approximately 63.21% of women fall within the height requirements, highlighting how normal distribution models can inform policy and standards.

Problem 3: Women’s pulse rates are normally distributed with a mean of 68.6 beats per minute (bpm) and a standard deviation of 10.9 bpm. The student is asked to find the z-score for a pulse rate of 66 bpm. The correction involves calculating:

z = (66 - 68.6) / 10.9 ≈ -0.24

This negative z-score indicates that 66 bpm is slightly below the average pulse rate. Understanding z-scores allows us to interpret how individual data points relate to the population mean and standard deviation, facilitating comparisons across different datasets.

Problem 4: The question asks for the cumulative area under the standard normal curve up to a z-score of -0.785 and the area on the right of this z-score. First, find the cumulative area:

Φ(-0.785) ≈ 0.2177

Since the total area under the curve is 1, the area to the right is:

1 - 0.2177 ≈ 0.7823

This demonstrates how to find areas on either side of a z-score using the symmetry and properties of the normal distribution.

Problem 5: Given the area of 0.6753 from the right, the z-score can be found via the inverse normal function:

z = NORM.INV(1 - 0.6753, 0, 1) ≈ 0.45

This indicates that 67.53% of the distribution lies to the right of the z-score of approximately 0.45, illustrating how to convert area probabilities into z-scores using statistical software functions.

Problem 6: The problem involves men’s shoulder widths with a mean of 18.2 inches and a standard deviation of 2.9 inches. To find the percentage of men fitting into a manhole of 22.5 inches in diameter, we calculate the z-score:

z = (22.5 - 18.2) / 2.9 ≈ 1.45

Using a z-table or calculator, Φ(1.45) ≈ 0.9265, meaning approximately 92.65% of men have shoulder widths less than 22.5 inches, hence will fit into the manhole. This application demonstrates how normal distribution helps in assessing product fit based on physical measurements.

Overall, these problems underscore the importance of understanding the properties of the standard normal distribution, using z-scores, and applying statistical tools like tables and software to derive probabilities and interpret data. Mastery of these techniques enables better decision-making across various fields, including healthcare, engineering, and business analytics.

References

  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. W.H. Freeman.
  • Agresti, A., & Finlay, B. (2009). Statistical Methods for the Social Sciences. Pearson.
  • Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis. Brooks/Cole.
  • Newbold, P., Carlson, W., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
  • Launer, J., & Wilkinson, D. (2011). Research Methods in Health: Investigating Health and Healthcare. Open University Press.
  • Walpole, R. E., Myers, R. H., & Myers, S. L. (2012). Probability & Statistics for Engineering and the Sciences. Pearson.
  • Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.
  • Heumann, M., & Shub, L. (2018). Statistics: The Art and Science of Learning from Data. Pearson.
  • Stock, J., & Watson, M. (2015). Introduction to Econometrics. Pearson.

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