The Heights Of The Adults In One Town Have A Mean Of 675 Inc

The Heights Of The Adults In One Town Have A Mean Of 675 Inches And A

The heights of the adults in one town have a mean of 67.5 inches and a standard deviation of 3.4 inches. Using Chebyshev's theorem, we can estimate the percentage of adults whose heights fall within a certain range around this mean. Specifically, we want to find the proportion of individuals with heights between 57.3 inches and 77.7 inches.

First, identify the mean (μ) and standard deviation (σ):

• Mean (μ) = 67.5 inches
• Standard deviation (σ) = 3.4 inches

Next, determine the distances of the interval bounds from the mean:

• Lower bound: 57.3 inches
• Upper bound: 77.7 inches

Calculate the deviation of each bound from the mean:

Lower deviation: 67.5 - 57.3 = 10.2 inches

Upper deviation: 77.7 - 67.5 = 10.2 inches

Both bounds are equally distant from the mean, at 10.2 inches, so the interval is symmetric around the mean.

Now, determine how many standard deviations this distance represents:

Number of standard deviations (k):

k = (Distance from mean) / σ = 10.2 / 3.4 ≈ 3

Thus, the interval from 57.3 inches to 77.7 inches covers approximately 3 standard deviations from the mean on both sides.

Applying Chebyshev's theorem, the theorem states that at least 1 - 1/k² of the data lies within k standard deviations of the mean, regardless of the data's distribution.

Substituting k = 3, we get:

Minimum percentage of adults within this range:

1 - 1/3² = 1 - 1/9 = 8/9 ≈ 0.8889 or 88.89%

Therefore, at least approximately 88.89% of the adults in the town are between 57.3 inches and 77.7 inches tall according to Chebyshev's theorem.

References

• Chebyshev, P. L. (1867). Sur la moyenne et la variance. Journal de Mathématiques Pures et Appliquées, 12, 177-184.
• Freund, J. E. (2010). Modern Elementary Statistics. Pearson Education.
• Ross, S. M. (2014). Introduction to Probability and Statistics. Academic Press.
• Wackerly, D. D., Mendenhall, W., & Scheaffer, R. L. (2008). Mathematical Statistics with Applications. Brooks/Cole.
• Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
• Moore, D. S., Notz, W., & Wildi, D. (2013). The Basic Practice of Statistics. W. H. Freeman.
• Kolmogorov, A. N. (1933). Foundations of the Theory of Probability. Chelsea Publishing Company.
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Paper For Above instruction

Understanding the distribution of human heights within a population is a fundamental aspect of biostatistics and demographic studies. In this context, the application of Chebyshev's theorem provides valuable insights into the variability and range of adult heights in a specific town. Given that the mean height is 67.5 inches with a standard deviation of 3.4 inches, employing Chebyshev's inequality allows us to determine the minimum proportion of the population falling within a specified interval without assuming any particular distribution shape.

Chebyshev's theorem states that for any distribution—regardless of its skewness or modality—the proportion of data within k standard deviations of the mean is at least 1 - 1/k². When the data points to a symmetric or bell-shaped distribution, the actual percentages are often higher; however, Chebyshev's theorem provides a conservative estimate applicable universally. In our case, the interval between 57.3 inches and 77.7 inches has been identified, and the task is to determine what percentage of adults have heights within this range.

Calculating the deviations, both bounds are 10.2 inches from the mean (since 67.5 - 57.3 = 10.2 and 77.7 - 67.5 = 10.2). When expressed in terms of standard deviations, this equates to 10.2 / 3.4 ≈ 3, implying that the interval spans approximately three standard deviations from the mean on either side. This symmetry ensures that the calculation is straightforward and applicable within Chebyshev's framework.

Applying Chebyshev's inequality with k = 3 yields a minimum of 1 - 1/9 = 8/9, or approximately 88.89%. This means that at least 88.89% of the adult population in the town has heights between 57.3 and 77.7 inches, providing a robust estimate that does not depend on the distribution's shape. The actual percentage may be higher, especially if the distribution is approximately normal, but Chebyshev's theorem guarantees this conservative lower bound.

Understanding the implications of this statistical bound is vital for public health planning, ergonomic design, clothing manufacturing, and health assessments within populations. Such estimations assist policymakers and industry stakeholders in making data-driven decisions, especially when the underlying distribution of heights is unknown or irregular.

In conclusion, Chebyshev's theorem serves as a powerful, distribution-independent tool to estimate population characteristics. Given the parameters provided, we confidently state that over 88.89% of adults in the town are within the specified height interval, illustrating the practical utility of this theorem in demographic analytics and statistical inference.