Problem 43 - Algebra and Trigonometry Real Mathematics, Real People

Question

The IQ scores for adults roughly follow the normal distribution \(y=0.0266 e^{-(x-100)^{2 / 450}}\) \(70 \leq x \leq 115,\) where \(x\) is the IQ score. (a) Use a graphing utility to graph the function. (b) Use the graph in part (a) to estimate the average IQ score.

Step-by-Step Solution

Answer

The average IQ score can be estimated by the peak of the normal distribution graph, which can be deduced from the graphing of the given function.

Step 1: Graph the Given Function

Utilize a graphing tool and input the provided function \(y=0.0266e^{-(x-100)^{2} / 450}\) making sure to set the domain as \(70 \leq x \leq 115\). Plotting this function will yield a bell curve which is characteristic of a normal distribution.

Step 2: Analyze the Graph

Inspect the graph visually to identify the highest point on the curve. This is the peak of the normal distribution, indicating the most frequent value.

Step 3: Estimate the Average IQ score

The x-value corresponding to the highest point on the curve represents the average IQ score in this distribution. This can be deduced because in a normal distribution, the peak of the curve or the mode corresponds to the average or mean score. Identify this x-value by approximation from the graph.

Key Concepts

Graphing Functions

Understanding how to graph functions is fundamental in visualizing mathematical concepts and interpreting data. In the context of normal distribution, graphing a function allows one to see the symmetrical bell-shaped curve that represents how data points are distributed.

When you use a graphing utility to plot a function such as the given IQ score function, you're essentially transforming the algebraic expression into a visual form. The function provided, which follows the equation

 \(y=0.0266 e^{-(x-100)^{2 / 450}}\) 

represents the distribution of IQ scores. By graphing this function within the domain of 70 to 115, we can visualize the spread and concentration of IQ scores across this range. The graphing process involves setting up axes, plotting points that satisfy the function, and connecting these points to form the characteristic curve.

In educational platforms, interactive tools are often provided to allow for dynamic graphing and better understanding of functions. Always remember to adjust your domain and range according to the context of the problem to ensure the entire relevant section of the curve is visible.

Normal Distribution Characteristics

The normal distribution, known for its bell shape, is a continuous probability distribution that is extremely important in statistics. Its characteristics help us understand phenomena that have a natural variability around an average value.

Key Characteristics of Normal Distribution

• Symmetry: The distribution is symmetric about the mean, meaning that the left half is a mirror image of the right half.
• Mean, Median, and Mode: In a perfect normal distribution, these three measures of central tendency are equal and located at the center of the distribution.
• Decay: The probabilities decay exponentially as one moves away from the mean, meaning occurrences become less likely the further they are from the average.
• Inflection Points: These occur at one standard deviation from the mean on either side, marking the change in curvature of the bell curve.

The graph function in the exercise represents a normal distribution where the mean IQ score can be visually identified since it corresponds with the peak of the curve. Also, since IQ scores follow this distribution, it implies that most people's IQs are around the average with fewer individuals having very low or very high scores.

Estimating Average from Graph

Estimating the average of a dataset from its graph is a crucial skill in interpreting data, particularly when analyzing normally distributed variables. With the graph of the normal distribution, estimating the average is straightforward because it's the peak of the curve.

In the graph of the IQ function

 \(y=0.0266 e^{-(x-100)^{2} / 450}\) 

for instance, you can visually scan for the highest point along the curve within the specified domain. For normally distributed data, this peak represents the mean or the 'average' IQ score. This peak is rightly centered in the graph because of the symmetric nature of the normal distribution.

In an educational setting, teachers often emphasize using tools like rulers or tracing with fingers to identify this peak accurately when a graphing utility is not available. It is important to note, however, that in practical applications, especially with empirical data, slight asymmetries might require more sophisticated methods to determine the mean accurately.