Problem 20 - Applied Calculus

Question

The percentage of people, \(P\), below the poverty level in the US " is given in Table 1.6. (a) Find a formula for the percentage in poverty as a linear function of time in years since \(2000 .\) (b) Use the formula to predict the percentage in poverty in 2006 (c) What is the difference between the prediction and the actual percentage, \(12.3 \% ?\) $$\begin{array}{c|c|c|c|c}\hline \text { Year (since 2000) } & 0 & 1 & 2 & 3 \\\\\hline P \text { (percentage) } & 11.3 & 11.7 & 12.1 & 12.5 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

(a) \( P(t) = 0.4t + 11.3 \), (b) 13.7%, (c) 1.4% higher.

Step 1: Determine a Linear Model

To model the percentage in poverty, we need to identify the slope and y-intercept for the linear equation of the form \( P(t) = mt + b \), where \( t \) is time in years since 2000 and \( P(t) \) is the percentage of people below the poverty level.Calculate the slope \( m \) using two data points from the table, say (year 0, 11.3) and (year 3, 12.5):\[m = \frac{12.5 - 11.3}{3 - 0} = \frac{1.2}{3} = 0.4.\]The line equation so far is \( P(t) = 0.4t + b \). To find \( b \), substitute one point from the table, e.g., (0, 11.3):\[11.3 = 0.4(0) + b \b = 11.3.\]Thus, the formula is \( P(t) = 0.4t + 11.3 \).

Step 2: Predict the Percentage in 2006

Use the formula \( P(t) = 0.4t + 11.3 \) to predict the percentage in poverty for the year 2006, which is \( t = 6 \):\[P(6) = 0.4(6) + 11.3 = 2.4 + 11.3 = 13.7.\]The predicted percentage for the year 2006 is 13.7%.

Step 3: Calculate the Difference with Actual Percentage

The actual percentage in 2006 is given as 12.3%. Find the difference between the predicted percentage and the actual percentage:\[\text{Difference} = 13.7 ext{ ext%} - 12.3 ext{ ext%} = 1.4 ext{ ext%}.\]The prediction was 1.4% higher than the actual percentage.

Key Concepts

Slope Calculation

When dealing with linear regression, one of the first steps is to determine the slope of the line. The slope is essentially the rate at which the percentage of people below the poverty line changes over time. It's an important aspect because it tells us how quickly or slowly the poverty percentage is increasing or decreasing.

To find the slope (\( m \)), we use the formula for the slope of a line between two points, which is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This calculation involves selecting two points from the data table. For example, we can use the points for the years since 2000: (0, 11.3) and (3, 12.5). Here '0' and '3' denote years since 2000 and '11.3' and '12.5' represent the poverty percentages in those years, respectively.

• Calculate the difference between the percentages: \( 12.5 - 11.3 = 1.2 \)
• Calculate the difference between the years: \( 3 - 0 = 3 \)
• Find the slope: \( m = \frac{1.2}{3} = 0.4 \)

The slope here is 0.4, which suggests that each year the percentage of people in poverty increases by 0.4%.

Predictive Modeling

Once we've calculated the slope and the y-intercept, we can use them to create a predictive model. Linear models are powerful because they allow us to forecast future values based on past trends.

The linear equation for our data is in the form \( P(t) = mt + b \), where \( P(t) \) is the poverty percentage at year \( t \), \( m \) is the slope, and \( b \) is the y-intercept which we previously calculated as 11.3.

• In our example, the formula becomes \( P(t) = 0.4t + 11.3 \)
• To predict the percentage in poverty for 2006, which is year 6 since 2000, we substitute \( t = 6 \)
• Perform the calculation: \( P(6) = 0.4 \times 6 + 11.3 = 13.7 \)

Using this formula, we predicted that in 2006 the poverty percentage would be 13.7%.

Percentage Difference Calculation

Comparing predictions with actual results is key to assessing the accuracy of our models. Understanding the percentage difference gives us insight into how closely our model's outputs match real-world measures.

The given actual poverty percentage for 2006 is 12.3%. We previously predicted it to be 13.7%. The difference can be determined by finding how much higher our prediction was from the actual percentage.

• Calculate the difference: \( 13.7 \text{\%} - 12.3 \text{\%} = 1.4 \text{\%} \)
• This reveals that our model overestimated the poverty percentage by 1.4% in 2006.

Thus, knowing the percentage difference helps us in refining our model and understanding any biases present in the predictive analysis.