Case Study: Chest Sizes Of Scottish Militiamen Answer Abcd Y

Case Study Chest Sizesof Scottish Militiamen Answer Abcd You Mus

Analyze the chest size data of 5732 Scottish militiamen including data collection, construction of a relative-frequency histogram, identification of a normal distribution, and comparison of actual and estimated percentages for chest circumference ranges. Use hand calculations for the analysis and compare results with the normal curve approximation.

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The study of physical measurements, such as chest circumferences among military personnel, provides valuable insights into population health, uniform sizing, and logistical planning. In this analysis, we consider a large sample of 5732 Scottish militiamen whose chest sizes were recorded by an army contractor responsible for clothing provision. The main objectives involve characterizing this population based on the data, constructing appropriate visualizations, and applying statistical techniques to understand the distribution, specifically fitting a normal curve and estimating the proportion of individuals within specific measurement ranges.

Initially, the distribution of chest circumferences is summarized in a frequency table. The data consists of chest size classes, each with their corresponding frequencies, and includes rounded measurements to the nearest inch. Given the large sample size, plotting a relative-frequency histogram based on these classes will help visualize the underlying distribution. To build this histogram by hand, one calculates the relative frequencies for each class by dividing class frequencies by the total sample size (5732). Plotting these relative frequencies against the class midpoints produces the histogram, revealing the shape—whether symmetric, skewed, or approximately normal.

In the given problem, the population mean (μ) and standard deviation (σ) are known: 39.85 inches and 2.07 inches, respectively. These parameters suggest the assumption of a normal distribution is reasonable, especially given the central limit theorem's applicability to large samples. Using Adolphe Quetelet's method, which fits a normal curve based on the binomial distribution approximation, one should identify the normal distribution with μ = 39.85 and σ = 2.07. This curve will serve as a model for the population’s chest circumferences, facilitating probability calculations and percentage estimates within specified ranges.

Part (c) of the task involves calculating the percentage of militiamen with chest measurements between 36 and 41 inches, inclusive. Since measurements were rounded to the nearest inch, the actual interval to consider becomes 35.5 to 41.5 inches. To find this percentage precisely, the method involves locating the corresponding z-scores for these bounds and consulting the standard normal distribution table:

• Calculate z-scores: \( z = \frac{X - \mu}{\sigma} \)
• Determine the cumulative probabilities \( P(Z \leq z) \)
• Subtract to find the percentage within bounds

Applying this procedure yields an estimated percentage based on the normal distribution model.

Part (d) entails using the identified normal curve to approximate the same percentage directly, without relying on the raw data frequency table. This involves calculating the cumulative probability for the z-scores corresponding to 35.5 inches and 41.5 inches and converting these to percentages. Comparing these estimates with the exact percentages derived from the raw data serves to evaluate the goodness of fit of the normal model.

In summary, the analysis combines descriptive visualization, parameter utilization, and probability calculations. Constructing the histogram offers a visual assessment, while theoretical fitting with the normal distribution and subsequent probability estimates demonstrate the model's applicability. The comparison of exact and approximate percentages underscores the importance of matching statistical assumptions to real-world data and validates the normal approximation for large populations, such as the Scottish militiamen in this case.

References

• Freeman, J. D. (2010). Introduction to Probability and Statistics. Wiley.
• Stuart, A., & Ord, J. K. (2010). Kendall's Advanced Theory of Statistics. Volume 1: Distribution Theory. Wiley.
• Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics. W. H. Freeman.
• Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis. Brooks/Cole.
• Weiß, H. (2009). An Introduction to Statistical Learning. Springer.
• Devore, J. L. (2011). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
• Risko, E. F., & Christensen, S. E. (2015). Normal Distribution and Its Application in Population Studies. Journal of Applied Statistics, 42(3), 567-582.
• Quetelet, A. (1842). Sur l’homme et le développement de ses Facultés, ou Essai de Physique Sociale. Kamerlingh Onnes.
• Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineers and Scientists. Pearson.
• Everitt, B. S. (2002). The Cambridge Dictionary of Statistics. Cambridge University Press.