Problem 37 - College Algebra and Calculus: An Applied Approach

Question

Women's Heights The distribution of heights of American women (between 30 and 39 years of age) can be approximated by the function \(p=0.163 e^{-(x-64.9)^{2} / 12.03}, \quad 60 \leq x \leq 74\) 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 women in this age bracket. (Source: U.S. National Center for Health Statistics)

Step-by-Step Solution

Answer

The average height of American women aged between 30 and 39 years is 64.9 inches.

Step 1: Analyzing the Function

The given function \(p=0.163 e^{-(x-64.9)^{2} / 12.03}, \quad 60 \leq x \leq 74\) is a Gaussian (or normal) distribution function, where the height \(x\) varies from 60 to 74 inches. The analysis starts with understanding the shape of the curve, which can be represented using a graphing utility.

Step 2: Graphing the Function

Use a graphing utility to plot the function \(p=0.163 e^{-(x-64.9)^{2} / 12.03}\). The graph will appear as a bell curve, symmetric about its peak. The peak of the curve indicates the most common (modal) height, which in this case is 64.9 inches.

Step 3: Determining the Average Height

In this symmetric distribution, the average (or mean) is equal to the most common (modal) height. Therefore, the average height of American women aged 30-39 is also 64.9 inches.

Key Concepts

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is one of the most important concepts in statistics. It is often depicted as a bell-shaped curve. This shape shows how data points are distributed in many real-world scenarios, including heights, test scores, and errors in measurements. The formula for a normal distribution includes several key parts, like the mean, variance, and standard deviation.

• The mean is the average or the most common value in the dataset, represented by the peak of the bell curve.
• Variance reflects how data points spread out from the mean, and how wide or narrow the bell curve appears.
• The standard deviation is the square root of the variance, indicating how much variation or dispersion exists from the mean.

In many cases, the normal distribution helps to predict probabilities and make decisions in uncertain conditions. It plays a critical role in statistics and data analysis, representing numerous independent variables as a central limit strategy.

Average Height

In the context of a normal distribution, finding the average height can be quite straightforward. Since the distribution is symmetric around its mean, the most common or modal value is often also the mean or average.

For example, in the given exercise about American women's heights, the mean or average height is 64.9 inches. This is because the peak of the normal distribution curve, which is the highest point, appears at 64.9. The symmetry ensures that the data is evenly spread around this point.

• The Mean: The sum of all values divided by the number of values, represents the central or typical value in a data set. For an ideal normal distribution, this value is the same as the mode and median.
• The Median: The middle number when we order the data set from least to greatest.
• The Mode: The value that appears most frequently in the data set.

In scenarios with symmetric normal distributions, these three measures (mean, median, and mode) coincide, giving a clear picture of the average.

Graphing Utility

A graphing utility is a powerful tool used to visualize mathematical functions. It transforms abstract equations into visual graphs that are easier to understand.

• Understanding Functions: By plotting equations such as the Gaussian distribution, one can see the bell-shaped curve representing the probability distribution.
• Identifying Key Points: Graphing utilities help locate the peak, or the highest point of the curve, which can be crucial for identifying the mode or average.
• Visualizing the Spread: You can observe the spread or dispersion of data, seeing how tightly or widely data points are clustered around the mean.

By using a graphing utility, students and professionals can better analyze, solve, and comprehend functions, especially when dealing with complicated equations. These tools can significantly enhance understanding, making abstract concepts like normal distribution more tangible for learners.