Question

Atmosphere The concentration \(y\) (in parts per million) of carbon dioxide in the atmosphere is measured at the Mauna Loa Observatory in Hawaii. The greatest monthly carbon dioxide concentrations for the years 2002 to 2006 are shown in the table. (Source: Scripps CO2 Program) $$ \begin{array}{|c|c|c|} \hline \text { Year } & t & \text { Concentration, } y \\ \hline 2002 & 0 & 375.55 \\ \hline 2003 & 1 & 378.35 \\ \hline 2004 & 2 & 380.63 \\ \hline 2005 & 3 & 382.26 \\ \hline 2006 & 4 & 384.92 \\ \hline \end{array} $$ (a) Solve the following system for \(a\) and \(b\) to find the least squares regression line \(y=a t+b\) for the data. Let \(t\) represent the year, with \(t=0\) corresponding to 2002 . \(\left\\{\begin{array}{r}5 b+10 a=1901.71 \\ 10 b+30 a=3826.07\end{array}\right.\) (b) Use a graphing utility to graph the regression line and predict the largest monthly carbon dioxide concentration in 2012 . (c) Use the regression feature of a graphing utility to find a linear model for the data. Compare this model with the one you found in part (a).

Short answer

The least squares regression line of the data is \(y = 2.265t + 375.61\). The predicted carbon dioxide concentration for the year 2012 is 398.26 parts per million. The comparison result will depend on the model given by the graphing utility, which can vary slightly due to computational accuracy.

Step-by-step solution

Solving the System of Linear Equations

This system is composed of two linear equations with two unknowns \(a\) and \(b\). There are multiple methods to solve such systems, one of which is substitution or elimination. Using elimination here seems to be easier for computation. Multiply the first equation by 2 and subtract it from the second one, resulting in: \(\displaystyle 10b + 30a - 2(5b + 10a) = 3826.07 - 2*1901.71\), simplifying we get: \(\displaystyle 10a = 22.65, a = 2.265\). Substituting \(a\) into the first equation: \(\displaystyle 5b + 10(2.265) = 1901.71 , 5b = 1901.71 - 22.65, b = 375.61\). So, the least squares regression line can be written as : \(y = 2.265t + 375.61\).

Graph the Regression Line and Predicting

Insert the calculated values of \(a\) and \(b\) into the linear function and graph the resulting line using a graphing utility. Upon graphing the function \(y = 2.265t + 375.61\), a prediction can be made by plugging in \(t = 10\) because the year we want (2012) is ten years from our model's starting point (2002). So, the prediction is \(y = 2.265*10 + 375.61 = 398.26\) parts per million.

Comparing with Graphing Utility Linear Regression Model

By inputting the data points into the graphing utility, the utility can generate a linear regression model. This can then be compared to the one generated manually. Differences can be attributed to potential computational accuracies provided by the utility.

Conclusion and Comparison

Closely compare this model to the one created in the previous steps to see how well the manual calculation aligns with the utility-generated prediction.

Key concepts

Linear Regression

Linear regression is a statistical method that helps us understand the relationship between two variables by fitting a linear equation to observe data. This technique is broadly used in forecasting and predicting trends, finding correlations, and creating models for analysis. The primary goal is to establish a line that best represents the data distribution.

For example, in the process of determining the atmospheric carbon dioxide concentration, linear regression helps in predicting future concentrations based on past data. When plotted, the data points may not perfectly align, which is where the regression line, often denoted as \( y = at + b \), comes into play. Here, \( y \) is the dependent variable, \( t \) is the independent variable (typically time), and \( a \) and \( b \) are the coefficients that determine the slope and intercept of the line.

Properly applying linear regression allows us to make informed predictions about the trend of the data, like estimating future CO2 levels in our initial problem. This can hold significant implications in environmental studies and climate change analysis.

System of Linear Equations

A system of linear equations comprises two or more linear equations with multiple variables. They can be solved simultaneously to find common points that satisfy all equations simultaneously. In our example, we have a system involving two equations:
\[\begin{align*}5b + 10a & = 1901.71, \10b + 30a & = 3826.07.\end{align*}\]

To solve this system, methods like substitution and elimination are typically employed. In this instance, the elimination method is advantageous due to its straightforward calculation process. By manipulating one equation to eliminate a variable when combined with another, we derive values for \(a\) and \(b\).

Such systems are foundational in identifying the regression equation parameters and essential in finding that best-fit line, forming the core of many applied mathematics and engineering solutions.

Least Squares Method

The least squares method is crucial in linear regression, ensuring that the derived line minimizes the sum of squared differences between observed and predicted values. It essentially helps derive the most accurate line possible by reducing the discrepancy between real data points and the line.

The technique works by determining the line that has the smallest possible sum of squared vertical distances (errors) from the data points to the line itself. Mathematically, it identifies values of \(a\) and \(b\) in the linear equation \(y = at + b\) such that the expression \( \sum_{i=1}^{n}(y_i - (a*t_i + b))^2 \) is minimized.

In the context of our carbon dioxide analysis, this method was pivotal in calculating the regression line \(y = 2.265t + 375.61\).
The least squares method is incredibly valuable in fields requiring precise data modeling and predictions, ensuring results are as close to the real-world data as possible.