The IQ scores for a sample of a class of returning adult...

The IQ scores for a sample of a class of returning adult students at a small northeastern college roughly follow the normal distribution $y=0.0266 e^{-(x-100)^{2} / 450}, \quad 70 \leq x \leq 115$ where $x$ is the IQ score. (a) Use a graphing utility to graph the function. (b) From the graph in part (a), estimate the average IQ score of an adult student.

The IQ scores for a sample of a class of returning adult students at a small northeastern college roughly follow the normal distribution
$y=0.0266 e^{-(x-100)^{2} / 450}, \quad 70 \leq x \leq 115$
where $x$ is the IQ score.
(a) Use a graphing utility to graph the function.
(b) From the graph in part (a), estimate the average IQ score of an adult student.

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its symmetric, bell-shaped curve. It is defined by two parameters: the mean, which determines the center of the distribution, and the variance (or standard deviation), which measures the spread of the data. This distribution is widely used in statistics due to the central limit theorem, which explains why many real-world phenomena tend to follow a normal pattern under certain conditions.

Probability Density Function (PDF)

A probability density function describes the likelihood of a random variable taking on a particular value. For a continuous distribution like the normal distribution, the PDF shows how probabilities are distributed across the values of the variable. The area under the curve within a certain interval represents the probability that the variable falls within that interval.

Graphing Functions Using Technology

Using a graphing utility to plot functions allows for a visual representation of mathematical relationships. Graphing tools help in understanding the behavior of a function, such as identifying its peak, symmetry, and spread. This visual approach is particularly useful for complex functions or when precision and immediate visual feedback are required.

Estimation of Central Tendency from Graphs

Estimating the average or central value from the graph of a distribution involves visually identifying the peak or center of symmetry. In symmetric distributions like the normal distribution, the average is at the center of the bell curve. This concept is fundamental in statistics as it provides a quick and intuitive assessment of where most data values are concentrated.

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