Deliverable 02 Worksheet Instructions: The Following 195991

Deliverable 02 Worksheet instructions: The following worksheet is shown

The following worksheet presents several problems involving normal distribution, Z-scores, and probabilities. Your task is to help a student by explaining how to solve each problem step-by-step in detail. For each problem, identify if it contains errors, partial work, or is blank, and provide the correct, complete solution with clear explanations of each step. Be sure to include relevant statistical concepts, formulas, and calculations, citing appropriate sources if necessary. The goal is to guide the student through understanding how to approach, compute, and interpret normal distribution problems correctly.

Paper For Above instruction

Introduction

Understanding normal distribution, calculating probabilities, and interpreting Z-scores are fundamental skills in statistics. They enable us to analyze data related to naturally occurring variables such as bone density, height, and pulse rates. This paper aims to systematically solve and explain each of the problems presented in the worksheet, demonstrating the methodology and reasoning behind statistical computations and interpretations.

Problem 1: Probability of Bone Density Scores Between -1.53 and 1.98

Given that bone density scores follow a standard normal distribution, we are asked to find the probability that a randomly selected subject’s score lies between -1.53 and 1.98. The student has utilized Excel to find the cumulative probabilities corresponding to these Z-scores: P(Z

To find the probability that the score lies between these two Z-scores, we subtract the cumulative probability at -1.53 from that at 1.98: P(-1.53

This means there is approximately a 91.31% chance that a subject’s bone density score falls between -1.53 and 1.98.

Problem 2: Percentage of Women Meeting Height Requirements

The U.S. Airforce requires women to be between 64 inches and 77 inches tall. Women's height is normally distributed with a mean (μ) of 65 inches and a standard deviation (σ) of 3.5 inches. To find the percentage of women meeting this criterion, we first calculate the Z-scores for 64 inches and 77 inches:

• Z for 64 inches: (64 - 65) / 3.5 ≈ -0.2857
• Z for 77 inches: (77 - 65) / 3.5 ≈ 3.4286

Using standard normal tables or a calculator, we find:

• P(Z
• P(Z

The percentage of women between 64 and 77 inches is then:

0.9997 - 0.3875 ≈ 0.6122, or approximately 61.22%.

This indicates that about 61% of women meet the height requirement.

Problem 3: Z-Score for a Pulse Rate of 66 Beats Per Minute

Given that women’s pulse rates are normally distributed with mean 69.4 bpm and standard deviation 11.3 bpm, the Z-score for a pulse rate of 66 bpm is calculated as:

Z = (X - μ) / σ = (66 - 69.4) / 11.3 ≈ -0.3018

Thus, the Z-score is approximately -0.302, indicating the pulse rate 66 bpm is just below the mean.

The student’s answer should be corrected to reflect this calculated value, and it should include the explanation of standardizing a value to a Z-score.

Problem 4: Cumulative Area and Area on the Right for Z = -0.875

The area under the curve to the left of Z = -0.875 can be found using standard normal distribution tables or a calculator:

P(Z

The area to the right of Z = -0.875 is simply 1 minus this value:

1 - 0.1908 = 0.8092.

These calculations imply that about 19.08% of the distribution lies below Z = -0.875, while approximately 80.92% lies above it.

Problem 5: Z-Score Corresponding to an Area of 0.6573 to the Right

The student correctly applied Excel's NORM.INV function, using an area of 0.6573, to find the Z-score:

=NORM.INV(1 - 0.6573, 0, 1) = NORM.INV(0.3427, 0, 1) ≈ -0.41.

Because the area is on the right, the Z-score corresponds to the negative of this value. The correction is that the Z-score associated with an area of 0.6573 on the right side of the mean (left area being 0.3427) is approximately -0.41, not 0.41.

Hence, the negative Z-score indicates the position on the distribution left of the mean, aligning with the area under the curve on the right side.

Problem 6: Men’s Shoulder Width and Manhole Diameter

With men’s shoulder width normally distributed with a mean of 18.2 inches and a standard deviation of 2.09 inches, we are asked to determine what percentage of men will fit into a 22.5-inch diameter manhole.

First, find the Z-score for 22.5 inches:

Z = (22.5 - 18.2) / 2.09 ≈ 4.36.

Using standard normal tables or a calculator, P(Z

Therefore, approximately 100% of men will fit into the manhole with a diameter of 22.5 inches.

This illustrates how, in practical terms, the manhole size is sufficient for the entire population’s shoulder widths based on the distribution.

Conclusion

These problems demonstrate the application of normal distribution concepts, including calculating probabilities, Z-scores, and areas under the curve. Correct interpretation of these calculations is essential for making informed decisions based on statistical data. Teachers and students alike should be familiar with standard normal tables, calculator functions, and the steps involved in standardizing data to accurately analyze real-world situations.

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. (2017). Introduction to the Practice of Statistics (9th ed.). W. H. Freeman.
• Ott, L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis (6th ed.). Brooks/Cole.
• Stats Workbook for Excel. (2020). Statistics Explained. https://statisticsexplained.com
• U.S. Census Bureau. (2020). Statistical Abstract of the United States.https://census.gov
• Johnson, R. A., & Wichern, D. W. (2014). Applied Multivariate Statistical Analysis (6th ed.). Pearson.
• Larson, R., & Farber, T. (2016). Elementary Statistics: Picturing the World (7th ed.). Pearson.
• Siegel, S., & Castellan, N. J. (1988). Nonparametric Statistics for the Behavioral Sciences. McGraw-Hill.
• Hogg, R., McKean, J., & Craig, A. (2013). Introduction to Mathematical Statistics (7th ed.). Pearson.
• Wackerly, D., Mendenhall, W., & Scheaffer, R. (2008). Mathematical Statistics with Applications (7th ed.). Cengage Learning.