Problem 80 - Calculus Single Variable

Question

The following table records several paired values of automobile mileage \((x)\) measured in thousands of miles and hydrocarbon emissions per mile ( \(y\) ) measured in grams $$ \begin{array}{|l|l|r|r|r|r|} \hline \boldsymbol{x} & 5.013 & 10.124 & 15.060 & 24.899 & 44.862 \\ \hline \boldsymbol{y} & 0.270 & 0.277 & 0.282 & 0.310 & 0.345 \\ \hline \end{array} $$ Plot these points. Determine the regression line through \((44.862,0.345) .\) If this pattern continues for higher mileage cars, about how many grams of hydrocarbons per mile would a car with 100,000 miles emit?

Step-by-Step Solution

Answer

A car with 100,000 miles is predicted to emit approximately 0.380 grams of hydrocarbons per mile.

Step 1: Plot the Data Points

Our first task is to plot the given pairs of values on a coordinate plane. The x-coordinate, which denotes mileage (in thousands), and the y-coordinate, which denotes hydrocarbon emissions (in grams), are paired as follows: (5.013, 0.270), (10.124, 0.277), (15.060, 0.282), (24.899, 0.310), and (44.862, 0.345). Use a graph to plot these points, with mileage on the x-axis and emissions on the y-axis.

Step 2: Choose the Regression Line Strategy

We will determine the equation of the regression line that best fits the given data. We use the point (44.862, 0.345) as one point on our line. To find a suitable line, we may use linear regression methods such as the least squares method to calculate the slope (m) and intercept (c) for the line of best fit in the form: \( y = mx + c \).

Step 3: Compute the Regression Coefficients

To calculate the slope (m) and y-intercept (c), use the formulas: \( m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \) and \( c = \frac{(\sum y) - m(\sum x)}{n} \), where n is the number of data points. Calculate these sums based on the given values: \( \sum x = 100.958, \sum y = 1.484, \sum xy = 251.2806, \sum x^2 = 2808.428 \).

Step 4: Plug Values to Find Slope and Intercept

Using the calculated sums, let's find values: \( m = \frac{5(251.2806) - (100.958)(1.484)}{5(2808.428) - (100.958)^2} \) and \( c = \frac{1.484 - m(100.958)}{5} \). Solving these, we find the approximate slope \( m \) and intercept \( c \) to plug into the regression line equation.

Step 5: Calculate the Line Equation

Using the obtained slope and intercept, the equation of our regression line is: \( y = mx + c \). Substitute the specific values of m and c determined in the previous step to find the full equation. This represents the line of best fit for the data.

Step 6: Predict Emissions for a 100,000-Mile Car

To predict the emissions for a car with 100,000 miles, substitute \( x = 100 \) (since the mileage is given in thousands) into the regression line equation: \( y = mx + c \). Solve this to find \( y \), which represents the number of grams of hydrocarbons emitted per mile by a car with 100,000 miles.

Key Concepts

Linear Regression

Imagine trying to predict a certain outcome based on some known data. Linear regression is a statistical technique used for exactly this purpose. It finds the straight line, or "best fit line," that best represents the relationship between two variables. This line allows you to make predictions about one variable based on another.

Linear regression involves two key components: the slope and the intercept. These components help define the relationship direction and strength. The method is widely used because it simplifies complex relationships to straight lines. When applying linear regression, remember it's essential to have some existing data between the two variables. This process involves mathematical calculations to ensure the line accurately describes the data points' trend. Once you have a regression line, you can use it to make reasonable predictions for new data.

Slope-Intercept Form

Understanding how to write the equation of a line in the slope-intercept form is crucial for linear regression. This form is expressed as: \[ y = mx + c \]

• \( m \): The slope of the line, telling you how steep or flat the line is. It describes how much \( y \) changes for a unit change in \( x \).
• \( c \): The y-intercept, or the point where the line crosses the y-axis.

Our goal is to use this formula to fit the best possible line through the given data points. Once the slope \( m \) and y-intercept \( c \) are calculated through techniques like the least squares method, they can be plugged back into this form.

This equation makes it easy to predict \( y \) for any given \( x \). It's called 'slope-intercept' because these two elements - slope and intercept - are key to defining the straight line.

Data Points

In the context of regression analysis, data points are the values you plot on a graph. Each point represents a pair of variables. Here, mileage in thousands is plotted along the x-axis and hydrocarbon emissions in grams on the y-axis.

When you plot these:

• (5.013, 0.270)
• (10.124, 0.277)
• (15.060, 0.282)
• (24.899, 0.310)
• (44.862, 0.345)

you can visually identify any trends or patterns: - Do emissions increase with mileage? - Is the relationship direct or inverse? For regression analysis, data points are crucial because they form the foundation of your analysis. From them, you derive a model describing the data mathematically without needing further visual representation. Ultimately, these allow for accurate predictions for unseen data.

Least Squares Method

The least squares method is like a tool that finds the best-fitting straight line through a set of data points. It works by minimizing the differences between the predicted and observed values.

Here's how it works:- Calculate the differences between each actual data point and the predicted data point on the line.- Square these differences to avoid negative values cancelling out.- Sum them all up. The aim is to have these squared differences as small as possible.By doing the above, we find the slope \( m \) and the intercept \( c \) of the line of best fit, forming the equation \( y = mx + c \).

This process ensures that the line minimizes the error in predicted values across the dataset. Thus, ensuring a reliable regression line is fitted, which can be used for future predictions.