Problem 85 - Applied Calculus

Question

BEHAVIORAL SCIENCES: Smoking and Education According to a study, the probability that a smoker will quit smoking increases with the smoker's educational level. The probability (expressed as a percent) that a smoker with \(x\) years of education will quit is approximately \(y=0.831 x^{2}-18.1 x+137.3\) (for \(10 \leq x \leq 16\) ). a. Graph this curve on the window [10,16] by [0,100] . b. Find the probability that a high school graduate smoker \((x=12)\) will quit. c. Find the probability that a college graduate smoker \((x=16)\) will quit.

Step-by-Step Solution

Answer

39.8% for a high school graduate and 60.4% for a college graduate.

Step 1: Understanding the Function

We are given a quadratic function that models the probability of a smoker quitting based on their years of education. The function is: \( y = 0.831x^2 - 18.1x + 137.3 \), where \( y \) is the probability percentage and \( x \) is the years of education ranging from 10 to 16.

Step 2: Graphing the Curve

To graph the function \( y = 0.831x^2 - 18.1x + 137.3 \) over the interval \([10, 16]\) and probability range \([0, 100]\), plot points for integer values of \( x \) in this range. Evaluate the function for each value to find the corresponding \( y \), and then plot these points on the graph. Connect the points with a smooth curve to visualize the quadratic function.

Step 3: Calculating Probability for High School Graduate

To find the probability that a high school graduate smoker (\( x = 12 \)) will quit, substitute \( x = 12 \) into the function: \( y = 0.831(12)^2 - 18.1(12) + 137.3 \). Calculate: \( y = 0.831(144) - 217.2 + 137.3 = 119.664 - 217.2 + 137.3 \). Thus, \( y = 39.764 \). Therefore, the probability for a high school graduate is approximately 39.8%.

Step 4: Calculating Probability for College Graduate

Now, substitute \( x = 16 \) for a college graduate into the function: \( y = 0.831(16)^2 - 18.1(16) + 137.3 \). Calculate: \( y = 0.831(256) - 289.6 + 137.3 = 212.736 - 289.6 + 137.3 \). Thus, \( y = 60.436 \). Therefore, the probability for a college graduate is approximately 60.4%.

Key Concepts

Probability

Probability is a concept often explored in behavioral sciences to predict the likelihood of certain events. In this exercise, we are dealing with the probability of a smoker quitting smoking, as influenced by their educational level. Probability is expressed as a percentage in this scenario, where a higher percentage indicates a higher chance of quitting.

Understanding how probability functions is crucial to interpreting data effectively:

• Probability ranges from 0% (impossible) to 100% (certain).
• In our exercise, the specific probability for smokers quitting is modeled by a quadratic equation, which helps illustrate changes based on the smoker's years of education.
• Probabilities can provide real-world insights, such as identifying behavioral trends across different educational levels.

By using a quadratic function, we can quantify this likelihood based on measurable factors.

Quadratic Function

Quadratic functions are expressions of the form \( y = ax^2 + bx + c \). They describe a parabolic graph that opens upwards (when \( a > 0 \)) or downwards (when \( a < 0 \)).

In the given exercise, the quadratic function \( y = 0.831x^2 - 18.1x + 137.3 \) is used to model the probability that a smoker will quit. Let's break down its significance:

• The term \( 0.831x^2 \) represents the parabolic component contributing to the curve's shape.
• The term \( -18.1x \) affects the linear trend component, influencing how steeply the probability changes with education.
• The constant term \( 137.3 \) shifts the entire curve vertically, establishing a baseline probability level.

By substituting different values of \( x \) (years of education) within the function, we can calculate specific probabilities. This showcases the quadratic function's power in modeling complex real-world situations.

Educational Level

Education is a pivotal factor in many health-related behaviors, including smoking cessation. In our exercise, the model uses educational level as an input to predict the likelihood of a smoker quitting.

Educational levels often correlate with health outcomes for several reasons:

• Individuals with higher education tend to have better access to healthcare resources and information.
• There is a stronger association between education and health awareness, leading to behavior conducive to quitting smoking.
• Educational environments can foster and support skills necessary for managing behaviors, such as perseverance and self-regulation.

By incorporating educational levels into the probability model, we can gain insights into how educational attainment influences health decisions.

Graphing Functions

Graphing functions allows us to visually interpret mathematical relationships, making it easier to analyze and understand patterns.

For this problem, we graph the function \( y = 0.831x^2 - 18.1x + 137.3 \) within the window \(10 \leq x \leq 16\) along the x-axis and \(0 \leq y \leq 100\) along the y-axis. Here’s how graphing helps:

• The visual representation makes it easier to identify trends and the overall shape of the function’s behavior: in this case, a parabola.
• It allows us to effectively compute precise estimates by identifying the y-value associated with specific x-values, like when \( x = 12 \) or \( x = 16 \).
• Graphs show how rapidly or slowly the probability changes with changes in education level.

By analyzing the graph, we can conclude general tendencies, such as how education influences the likelihood of quitting smoking in a population.