Problem 71 - Applied Calculus

Question

Based on a recent study, the probability that someone is a smoker decreases with the person's income. If someone's family income is \(x\) thousand dollars, then the probability (expressed as a percentage) that the person smokes is approximately \(y=-0.31 x+40\) (for \(10 \leq x \leq 100)\) a. Graph this line on the window [0,100] by [0,50] . b. What is the probability that a person with a family income of \(\$ 40,000\) is a smoker? [Hint: Since \(x\) is in thousands of dollars, what \(x\) -value corresponds to \(\$ 40,000 ?]\) c. What is the probability that a person with a family income of \(\$ 70,000\) is a smoker? Round your answers to the nearest percent.

Step-by-Step Solution

Answer

Plot points (10, 36.9) and (100, 9); Probability for $40,000 is 28%, for $70,000 is 18%.

Step 1: Graphing the Line

To graph the line, identify key points from the equation \( y = -0.31x + 40 \). To do this, choose values of \( x \) within the range \( [10, 100] \) and calculate the corresponding \( y \)-values.- When \( x = 10 \): \[ y = -0.31 \times 10 + 40 = 36.9 \]- When \( x = 100 \): \[ y = -0.31 \times 100 + 40 = 9 \]Plot these points: (10, 36.9) and (100, 9) on the graph window \([0, 100] \) by \([0, 50] \) and draw the line connecting these points.

Step 2: Calculate Probability for $40,000 Income

Determine the \( x \)-value for a family income of \\( 40,000. Since \( x \) is in thousands of dollars, \\) 40,000 corresponds to \( x = 40 \).Use the line equation \( y = -0.31x + 40 \) to find the probability that someone with an income of \$ 40,000 is a smoker:\[ y = -0.31 \times 40 + 40 = 27.6 \]So, the probability is approximately 28\% when rounded to the nearest percent.

Step 3: Calculate Probability for $70,000 Income

Determine the \( x \)-value for a family income of \\( 70,000. Here, \\) 70,000 corresponds to \( x = 70 \).Use the line equation \( y = -0.31x + 40 \) to find the probability:\[ y = -0.31 \times 70 + 40 = 18.3 \]So, the probability is approximately 18\% when rounded to the nearest percent.

Key Concepts

Linear equations

Linear equations are mathematical expressions that represent a straight line when graphed. These equations are typically in the form of \( y = mx + b \), where:

• \( y \) is the dependent variable or output.
• \( m \) is the slope, indicating the line's steepness and direction.
• \( x \) is the independent variable or input.
• \( b \) is the y-intercept, showing where the line crosses the y-axis.

In the context of the exercise, the equation \( y = -0.31x + 40 \) suggests that the probability someone smokes decreases linearly as income increases. The slope \( -0.31 \) reflects the rate of this decrease, meaning for every \( 1,000 \) dollar increase in family income, the probability someone smokes falls by \( 0.31 \) percent.
To solve problems using linear equations, you often need to find specific output values by substituting given inputs or vice versa. By solving, you can understand how changes in one variable impact another.

Graphing techniques

Graphing techniques for linear equations involve translating an equation into a visual representation on a coordinate plane. The basic steps include:

• Selecting key points. For \( y = -0.31x + 40 \), chose values like \( x = 10 \) and \( x = 100 \) to see how \( y \) changes.
• Calculating \( y \)-values. At \( x = 10 \), \( y = 36.9 \) and at \( x = 100 \), \( y = 9 \).
• Plotting these points on a graph. Use a graph window that appropriately includes these points, in this case, \([0, 100] \) along the x-axis and \([0, 50] \) along the y-axis.
• Drawing a line through the points to visualize the relationship.

This technique is essential in understanding how variables interact and helps visualize real-world data, such as the probability of smoking relative to income, offering a clear picture of the correlation indicated by the equation.

Income analysis in statistics

Income analysis in statistics often explores how income levels affect behavior and decisions, like smoking habits in this case. By analyzing the relation via the provided linear equation, statistical tools can:

• Identify patterns. As income increases, smoking probability falls, suggesting a negative correlation.
• Predict outcomes. The equation can predict smoking probability, showing a likelihood of 28% for \( \\(40,000 \) income and 18% for \( \\)70,000 \).
• Make informed decisions. Understanding this relation can inform public health policy, such as targeting smoking cessation programs in lower-income areas.

These analyses help showcase how variables correlate and how one variable, like income, can be a significant factor in predicting another, such as lifestyle choices. Income analysis is invaluable for tapping into socioeconomic behaviors and improving societal outcomes.