Mat 3001 From The Time Of Early Studies By Sir Francis Galto

Question

Mat 3001from The Time Of Early Studies By Sir Francis Galton In the La Mat 3001from the time of early studies by Sir Francis Galton in the late nineteenth century linking it with mental ability, the cranial capacity of the human skull has played an important role in arguments about IQ, racial differences, and evolution, sometimes with serious consequences. (See, for example, S.J. Gould, "The Mismeasure of Man," 1996.) Suppose that the mean cranial capacity measurement for modern, adult males is 1174cc (cubic centimeters) and that the standard deviation is 239 cc. Complete the following statements about the distribution of cranial capacity measurements for modern, adult males. (a) According to Chebyshev's theorem, at least ____% of the measurements lie between ____ cc and ____ cc. (b) According to Chebyshev's theorem, at least ____% of the measurements lie between ____ cc and ____ cc. (c) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately ____% of the measurements lie between ____ cc and ____ cc. (d) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately ____% of the measurements lie between ____ cc and ____ cc.

Dr. Jack HW Helper · Accepted Answer

Introduction The study of cranial capacity has historically been intertwined with various hypotheses concerning intelligence, race, and evolution. Since the late nineteenth century, researchers like Sir Francis Galton have examined cranial measurements as potential indicators of mental ability, fueling debates that have profound scientific and social implications. Modern statistical concepts such as Chebyshev's theorem and the empirical rule allow us to analyze the distribution of cranial capacity measurements and understand the variability within the population. This paper applies these statistical tools to interpret the distribution of cranial capacity among modern adult males, providing insight into the range and proportion of measurements within specified intervals. Understanding Distribution and Statistical Theories Chebyshev's theorem is a fundamental statistical principle that applies to all data distributions, regardless of shape. It states that for any dataset with a finite mean and standard deviation, a certain minimum percentage of data points lie within a specific number of standard deviations from the mean. The empirical rule, on the other hand, is applicable specifically to bell-shaped, normal distributions, providing more precise probability intervals. Data Context and Parameters According to recent studies, the mean cranial capacity of modern adult males is approximately 1174 cc, with a standard deviation of 239 cc. These parameters enable the calculation of specific intervals using theorems and rules mentioned above to estimate the spread and concentration of data points. Application of Chebyshev’s Theorem a) To find the percentage of measurements within a certain range using Chebyshev’s theorem, we set the interval at a distance of two standard deviations from the mean. - The lower bound: 1174 - 2 * 239 = 1174 - 478 = 696 cc - The upper bound: 1174 + 2 * 239 = 1174 + 478 = 1652 cc Chebyshev's theorem guarantees that at least $1 - \frac{1}{k^2}$ o

Mat 3001 From The Time Of Early Studies By Sir Francis Galto

Mat 3001from The Time Of Early Studies By Sir Francis Galton In the La

Paper For Above instruction

Introduction

Understanding Distribution and Statistical Theories

Data Context and Parameters

Application of Chebyshev’s Theorem

Application of the Empirical Rule

Discussion

Conclusion

References