Problem 39 - Precalculus with Limits

Question

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.

Step-by-Step Solution

Answer

The average IQ score can be estimated directly from the graph. It should be close to 100, which is the mean of this normal distribution based on its formula.

Step 1: Interpret the function and Use a Graphing Utility

The given function is the equation of a probability function for a normal distribution. This form is used to model phenomena in physics, social science, and many other areas where the obtained data is around a central value with no bias to the left or right, and it gets close to the mean. Given the IQ scores, we can use any graphing utility software or online tool, like 'Graphical Calculators' or 'Desmos' to plot the function.

Step 2: Plotting the graph

To plot the graph, input the function \(y=0.0266 e^{-(x-100)^{2} / 450}\) into the graphing tool. Make sure to bound the value of x between 70 and 115 to visualize the distribution of IQ scores in the exact interval. The 'x' in the function represents the IQ score.

Step 3: Estimating the average

The graph of the function is a bell-shaped curve symmetry about a vertical line, also known as the bell curve. The highest point (the peak) of a normal distribution graph corresponds to its mean. From the graph, visually inspect where the peak of the curve is. This point represents the most frequent score and in a normal distribution, it's also the mean (average) IQ score. You estimate this value by looking at the 'x' coordinate of the peak.

Key Concepts

Graphing Utility

A graphing utility is a valuable tool that helps us visualize mathematical functions and equations. In this case, it enables us to plot the given probability function of the normal distribution for IQ scores. There are various graphing utilities available, such as:

• Online tools like Desmos.


• Graphing calculators including TI-84 or similar devices.


• Software programs like GeoGebra or MATLAB.

To start, input the function, in this case, \(y = 0.0266 e^{-(x-100)^{2} / 450}\), into the graphing utility. It is important to limit the value of \(x\) between 70 and 115, as specified in the problem. This ensures we correctly represent the distribution of IQ scores within the given range. Once the function is entered and the range is set, you will see a bell-shaped curve, providing a clear visualization of how IQ scores are distributed among students.

Average IQ Score

The average IQ score can be gleaned from the graph produced by a graphing utility. A normal distribution's characteristic bell curve reveals that the mean, or average, is located at the peak of the curve. In our scenario, once the function \(y = 0.0266 e^{-(x-100)^{2} / 450}\) is plotted, the highest point on the graph represents this average value of the students' IQ scores.

When examining the graph, look for the point where the bell curve's peak resides on the x-axis. This point is crucial because it shows the most common IQ score among the adult students. In a normal distribution, this mode is also the median and the mean, making it easy to estimate the average IQ score directly from the visualized graph. For this function, with the curve centered roughly around 100, it suggests that this is the approximate average IQ score of the group.

Probability Function

The probability function in the context of the normal distribution, such as \(y = 0.0266 e^{-(x-100)^{2} / 450}\), is a mathematical representation that describes how data is dispersed in relation to the mean. This particular function represents a bell curve that is symmetric around the mean (100 in this case).

In general, the probability function is key in determining the likelihood of obtaining a particular value within a dataset. For IQ scores, this means understanding what proportion of scores fall within certain ranges or how likely it is for someone to have a score close to the mean.

• The mean determines the central peak of the curve.


• The standard deviation, indirectly influenced by numbers like 450 in the given formula, reflects the spread or width of the bell curve.

This smooth, continuous curve helps predict probabilities and analyze data distributions, making it a fundamental concept for fields that rely on statistical analysis.