Deliverable 02 Worksheet Instructions 258453

Deliverable 02 Worksheetinstructions the Following Worksheet Is Show

The following worksheet is shown to you by a student who is asking for help. Your job is to help the student walk through the problems by showing the student how to solve each problem in detail. You are expected to explain all of the steps in your own words. Key: · - This problem is an incorrect. Your job is to find the errors, correct the errors, and explain what they did wrong. ·

- This problem is partially finished.

You must complete the problem by showing all steps while explaining yourself. · - This problem is blank. You must start from scratch and explain how you will approach the problem, how you solve it, and explain why you took each step.

  1. Assume that a randomly selected subject is given a bone density test. Those tests follow a standard normal distribution. Find the probability that the bone density score for this subject is between -1.63 and 1.96. Student’s answer: We first need to find the probability for each of these z-scores using Excel. For -1.63 the probability from the left is 0.0516, and for 1.96 the probability from the left is 0.9750. Continue the solution: Finish the problem giving step-by-step instructions and explanations.

  2. The U.S. Airforce requires that pilots have a height between 64 in. and 77 in. If women’s heights are normally distributed with a mean of 65.2 in. and a standard deviation of 3.53 in, find the percentage of women that meet the height requirement. Answer and Explanation: Enter your step-by-step answer and explanations here.

  3. Women’s pulse rates are normally distributed with a mean of 68.6 beats per minute and a standard deviation of 10.9 beats per minute. What is the z-score for a woman having a pulse rate of 66 beats per minute? Student’s answer: Let Corrections: Enter your corrections and explanations here.

  4. What is the cumulative area from the left under the curve for a z-score of -0.785? What is the area on the right of that z-score? Answer and Explanation: Enter your step-by-step answer and explanations here.

  5. If the area under the standard normal distribution curve is 0.6753 from the right, what is the corresponding z-score? Student’s answer: We plug in “=NORM.INV(0.6753, 0, 1)” into Excel and get a z-score of 0.45. Corrections: Enter your corrections and explanations here.

  6. Manhole covers must be a minimum of 22 in. in diameter, but can be as much as 60 in. Men have shoulder widths that are normally distributed with a mean of 18.2 and a standard deviation of 2.9 in. Assume that a manhole cover is constructed with a diameter of 22.5 in. What percentage of men will fit into a manhole with this diameter? Student’s answer: We need to find the probability that men will fit into the manhole. The first step is to find the probability that the men’s shoulder is less than 22.5 inches. Continue the solution: Enter your step-by-step answer and explanations here.

Paper For Above instruction

This paper addresses the statistical analysis of normally distributed variables, focusing on z-scores, probabilities, and percentage calculations relevant to real-world scenarios such as medical testing, military standards, and engineering constraints.

Introduction

Understanding the properties of the standard normal distribution is essential in many fields, including healthcare, military, and engineering. This paper demonstrates the step-by-step problem-solving process, highlighting the importance of z-scores, cumulative probabilities, and the application of Excel functions to compute necessary statistical measures.

Problem 1: Bone Density Test Probability

The first problem involves calculating the probability that a randomly tested subject’s bone density score falls between -1.63 and 1.96 on the standard normal distribution. Using the empirical rule and z-table or Excel functions, we find the probabilities associated with each z-score.

From Excel’s NORM.S.DIST function, the cumulative probability for -1.63 is approximately 0.0516 and for 1.96 is approximately 0.9750. To find the probability that a score is between these two z-scores, we subtract the lower cumulative probability from the higher one:

P(-1.63

Therefore, there is a 92.34% chance that a subject’s bone density score falls between -1.63 and 1.96.

Problem 2: Women's Height Percentage

The second problem considers women's heights modeled by a normal distribution with a mean of 65.2 inches and a standard deviation of 3.53 inches. The goal is to find the percentage of women whose heights fall between 64 inches and 77 inches.

Calculate the z-scores for 64 and 77 inches:

  • For 64 inches: z = (64 - 65.2) / 3.53 ≈ -0.34
  • For 77 inches: z = (77 - 65.2) / 3.53 ≈ 3.50

Using Excel’s NORM.S.DIST function, the cumulative probabilities are:

  • P(Z
  • P(Z

The percentage of women within this height range is:

(0.9998 - 0.3669) * 100 ≈ 63.29%

Thus, approximately 63.29% of women meet the height requirements.

Problem 3: Pulse Rate Z-Score

The third problem asks for the z-score of a woman with a pulse rate of 66 beats per minute, given the normal distribution with a mean of 68.6 and a standard deviation of 10.9.

Using the formula for z-score:

Z = (X - μ) / σ = (66 - 68.6) / 10.9 ≈ -0.239

Hence, the z-score corresponding to a pulse rate of 66 bpm is approximately -0.239.

This indicates that a pulse rate of 66 bpm is slightly below the mean.

Problem 4: Area Under the Curve for Z = -0.785

To find the cumulative area from the left under the curve for Z = -0.785, we use Excel’s NORM.S.DIST function:

P(Z

The area to the right of Z = -0.785 is:

1 - 0.2164 = 0.7836.

This demonstrates that approximately 21.64% of the distribution lies to the left, and 78.36% to the right of Z = -0.785.

Problem 5: Z-Score from Given Area

Given the area of 0.6753 to the right, the area to the left is 1 - 0.6753 = 0.3247. Using Excel’s NORM.S.INV function:

Z = NORM.S.INV(0.3247) ≈ -0.39

The earlier answer of 0.45 is incorrect because it corresponds to a different cumulative area. The correct z-score for the area 0.3247 is approximately -0.39.

Problem 6: Men’s Shoulder Width and Manhole Diameter

The last problem involves calculating the percentage of men who can fit into a 22.5-inch diameter manhole, assuming shoulder width is normally distributed with a mean of 18.2 inches and a standard deviation of 2.9 inches.

Calculate the z-score:

Z = (22.5 - 18.2) / 2.9 ≈ 1.45

Using Excel’s NORM.S.DIST:

P(Z

Therefore, approximately 92.65% of men have shoulder widths less than 22.5 inches and can fit through the manhole.

Conclusion

Mastering the use of z-scores and the standard normal distribution is fundamental for interpreting probabilities in various applications. Using tools like Excel enhances accuracy and efficiency in these calculations. Understanding these concepts allows for practical assessment of probabilities relevant to health, safety, and engineering standards.

References

  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (8th ed.). Cengage Learning.
  • Mendenhall, W., Sincich, T., & Sorensen, K. (2012). A First Course in Statistics (10th ed.). Pearson.
  • Ott, L. (2012). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.
  • Behr, J. G. (2017). Statistics: Principles and Methods. Duxbury Press.
  • NIST/SEMATECH. (2012). e-Handbook of Statistical Methods. National Institute of Standards and Technology.
  • Microsoft. (2020). NORM.S.DIST and NORM.S.INV functions documentation. Microsoft Support.
  • Moore, D. S., Notz, W. I., & Falk, J. (2014). The Basic Practice of Statistics (7th ed.). W. H. Freeman.
  • Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
  • Wackerly, D. D., Mendenhall, W., & Scheaffer, R. L. (2008). Mathematical Statistics with Applications. Brooks/Cole.
  • Kruskal, W. H., & Wallis, W. A. (1952). Use of ranks in one-criterion variance analysis. Journal of the American Statistical Association, 47(260), 583-621.

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