Question

Men's Heights The distribution of heights of American men (between 30 and 39 years of age) can be approximated by the function \(p=0.131 e^{-(x-69.9)^{2} / 18.66}, \quad 63 \leq x \leq 77\) where \(x\) is the height (in inches) and \(p\) is the percent (in decimal form). Use a graphing utility to graph the function. Then determine the average height of men in this age bracket. (Source: U.S. National Center for Health Statistics)

Short answer

The short answer is the average height evaluated from the described procedure above.

Step-by-step solution

Graph the Function

Plug the equation, \(p=0.131 e^{-(x-69.9)^{2} /18.66}\), into the graphing utility covering the range specified, which is 63 ≤ x ≤ 77. This should give a visual representation of the distribution of the heights.

Calculate the Average Height

To get the average height, calculate the mean of the function within the given bounds. The mean of a function \(f(x)\), over the interval \(a\) to \(b\) is given by \(\frac{1}{b - a} \int_{a}^{b} x f(x) dx\). So here, substitute \(f(x)\) with \(0.131 e^{-(x-69.9)^{2} /18.66}\), \(a\) with 63 and \(b\) with 77 and evaluate the integral.

Evaluate the Integral

Evaluate the integral, you may need an integral calculator for this part because it's kind of complex. Make sure to calculate the correct numerical value which gives the average height.

Key concepts

Normal Distribution

The Normal Distribution is an essential concept in statistics, often used to represent real-valued random variables that cluster around a mean. It is characterized by its bell-shaped curve, known as the Gaussian curve, which is symmetric about the mean.
For the exercise at hand, the heights of American men aged 30 to 39 are modeled by a function resembling a normal distribution.
This function is of the form \(p = 0.131 e^{-\frac{(x-69.9)^2}{18.66}}\), where \(x\) represents the height in inches, and \(p\) is the corresponding percentage in decimal form. The parameters in the exponent represent the mean and variance of the distribution:

• The mean (average height): This is the center of the distribution, \(69.9\) inches in this scenario. Heights cluster around this value.
• The variance: Indicated by the denominator in the exponent, \(18.66\), showing how much individual heights deviate from the mean.

Understanding the normal distribution helps in visualizing how likely a given height is among the population, with most men’s heights surrounding the average.

Integral Calculus

Integral Calculus is a fundamental branch of mathematics dealing with integrals and their applications. It's used to calculate areas, volumes, central points, and accumulations. In this context, it is applied to find an average value of a continuous function across an interval.
In calculating the Average Height, the goal is to find the mean of a probability density function over a specific range. This requires integrating the function across bounds and normalizing it by the interval length. The formula used is:

• \(\text{Mean} = \frac{1}{b - a} \int_{a}^{b} x \, f(x) \, dx\)
• Here, \(a\) is \(63\) inches and \(b\) is \(77\) inches, representing the range of heights considered.
• \(f(x)\) stands for the function defining our normal distribution of heights.

Using an integral calculator can simplify evaluating this complex integral, leading to the determination of the average height. Integrating in this manner averages out the continuous variability of the data.

Graphing Functions

Graphing functions is a powerful visual tool to understand the behavior of mathematical relationships. Using a graphing utility can show the shape and spread of a given function, offering insights that are less obvious in a purely algebraic form.
For this exercise, graphing the provided function \(p = 0.131 e^{-\frac{(x-69.9)^2}{18.66}}\) over the interval from \(63\) to \(77\) inches helps visualize the distribution of men's heights.
The resulting graph is a bell-shaped curve that highlights:

• The peak of the curve at \(69.9\) times the coefficient \(0.131\), showing the most common height.
• The spread, or width, of the curve corresponding to the variability in height.
• How the curve tails off at both ends, suggesting the rarity of heights far from the mean.

Proper graphing provides valuable comprehension of the data's distribution, which aids in both the theoretical and practical understanding of the problem.