Problem 40 - Precalculus

Question

Suppose the final exam in this class has a normal, or bell-shaped, grade distribution of exam scores, with an average score of \(80 .\) An approximate function that models your class's grades on the exam is \(N(x)=10 e^{-(x-80)^{2} / 16^{2}},\) where \(N\) represents the number of students who received the score \(x\) a. Graph this function. b. What is the average grade? c. Approximately how many students scored a \(60 ?\) d. Approximately how many students scored \(100 ?\)

Step-by-Step Solution

Answer

The average grade is 80. Approximately 2 students scored 60, and about 2 scored 100.

Step 1: Graphing the Function

To graph the function \( N(x) = 10 e^{-(x-80)^2 / 16^2} \), we note that this is a Gaussian function, centered at 80 with a standard deviation related to 16. You can use a graphing calculator or software like Desmos or GeoGebra to plot it. Enter the function, adjust the x-axis to cover scores from 0 to 100, and observe the bell shape.

Step 2: Identify the Average Grade

The function is centered at \( x = 80 \), which corresponds to the mean of the distribution. In a normal distribution, this center point is the average. Thus, the average grade is simply 80.

Step 3: Calculating Students Scoring 60

To calculate the number of students who scored 60, substitute \(x = 60\) into \(N(x)\): \[N(60) = 10 e^{-(60-80)^2 / 16^2}\] Calculate \[(60-80)^2 = 400 \quad \text{and} \quad 16^2 = 256\]Now, \[N(60) = 10 e^{-400/256} \approx 10 e^{-1.5625} \approx 10 \times 0.2096 \approx 2.096\]Approximately 2 students scored around 60.

Step 4: Calculating Students Scoring 100

Substitute \(x = 100\) into \(N(x)\):\[N(100) = 10 e^{-(100-80)^2 / 16^2}\]Calculate \[(100-80)^2 = 400 \quad \text{and} \quad 16^2 = 256\]Now, \[N(100) = 10 e^{-400/256} \approx 10 e^{-1.5625} \approx 10 \times 0.2096 \approx 2.096\]Approximately 2 students scored around 100.

Key Concepts

Gaussian function

The Gaussian function, often referred to as a normal distribution, is a fundamental concept in statistics and probability. This function forms the basis for the normal distribution, which is a continuous probability distribution characterized by a symmetric, bell-shaped curve. In mathematics, the Gaussian function is expressed as:\[ N(x) = a \cdot e^{-(x-b)^2 / 2c^2} \]Here:

• \(a\) is the height of the curve's peak.
• \(b\) is the mean (or average) of the distribution, indicating where the peak is centered.
• \(c^2\) is related to the variance, determining the width of the distribution.

To graph a Gaussian function, you plot it over a range of values for \(x\), observing that most data points lie close to the mean value, and that the frequency of occurrence tapers off symmetrically as you move away from the mean.

average grade

In the context of a normal distribution, the average grade is also known as the mean. It is the central point around which the distribution is symmetrically positioned. For the exercise given, this is calculated directly from the function's expression.
The formula provided, \(N(x)=10e^{-(x-80)^2 / 16^2}\), tells us that the mean is at 80. This is because the exponent term \(x-b\) has \(b=80\), indicating that the curve's peak, which represents the most frequently occurring grade, is located at 80.
The average or mean is a crucial statistical measure, as it depicts the central tendency of the data set. In exams or grading, it helps educators understand students' overall performance and how it compares to the expected standard.

graphing calculator

A graphing calculator is an essential tool for students and educators dealing with mathematical functions like the Gaussian function. It allows the user to input complex equations and see visual representations of these functions.
For the problem at hand, you use the graphing calculator to visualize the normal distribution of grades given by the equation \(N(x)=10e^{-(x-80)^2 / 16^2}\). By plotting this equation, it helps students to easily identify the shape and spread of the distribution—a bell curve centered around the mean.

• Input the function into the calculator.
• Adjust the x-axis to encompass values from 0 to 100 to cover potential exam scores.
• Press 'Graph' to see the bell-shaped curve.

This kind of visual aid is invaluable, as it aids in comprehending abstract statistical concepts and their real-world applications.

bell-shaped curve

The bell-shaped curve is a visual representation of a Gaussian distribution. It is symmetric around its center, and its shape indicates that most occurrences of a value are concentrated around the mean, with probabilities tapering off as one moves away from the center.
In the described exercise, the bell-shaped curve centered at 80 shows that most students scored near this average. The curve's width is influenced by the standard deviation, which is related to the value \(16\) in the formula.

• The highest point of the curve represents the mean (average grade) of the distribution.
• The tails on each side represent lower-frequency scores, decreasing as they move away from the mean.

Understanding the properties of the bell-shaped curve helps predict patterns within data, making it a cornerstone concept in probability and statistics.