Question

Levels of carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) in the atmosphere are rising rapidly, far above any levels ever before recorded. Levels were around 278 parts per million in 1800 , before the Industrial Age, and had never, in the hundreds of thousands of years before that, gone above 300 ppm. Levels are now over 400 ppm. Table 2.31 shows the rapid rise of \(\mathrm{CO}_{2}\) concentrations over the 50 years from \(1960-2010\), also available in CarbonDioxide. \(^{73}\) We can use this information to predict \(\mathrm{CO}_{2}\) levels in different years. (a) What is the explanatory variable? What is the response variable? (b) Draw a scatterplot of the data. Does there appear to be a linear relationship in the data? (c) Use technology to find the correlation between year and \(\mathrm{CO}_{2}\) levels. Does the value of the correlation support your answer to part (b)? (d) Use technology to calculate the regression line to predict \(\mathrm{CO}_{2}\) from year. (e) Interpret the slope of the regression line, in terms of carbon dioxide concentrations. (f) What is the intercept of the line? Does it make sense in context? Why or why not? (g) Use the regression line to predict the \(\mathrm{CO}_{2}\) level in \(2003 .\) In \(2020 .\) (h) Find the residual for 2010 . Table 2.31 Concentration of carbon dioxide in the atmosphere $$\begin{array}{lc}\hline \text { Year } & \mathrm{CO}_{2} \\ \hline 1960 & 316.91 \\ 1965 & 320.04 \\\1970 & 325.68 \\ 1975 & 331.08 \\\1980 & 338.68 \\\1985 & 345.87 \\\1990 & 354.16 \\ 1995 & 360.62 \\\2000 & 369.40 \\ 2005 & 379.76 \\\2010 & 389.78 \\ \hline\end{array}$$

Short answer

The analysis indicates a strong, positive relationship between year and CO2 concentration, indicating that the latter has been continuously increasing. The slope of the regression line indicates the rate of growth in CO2 levels for every unit (year) increase. The intercept, however, doesn't have meaningful interpretation in this problem. The regression line can be used to predict future CO2 levels, though the error of these predictions can be quantified using residuals.

Step-by-step solution

Identify Variables

The explanatory variable is 'Year', which is used to predict the response variable 'CO2 Concentration'.

Draw Scatterplot

Plot the given data in a scatterplot, with 'Year' on the x-axis and 'CO2 level' on the y-axis.

Find Correlation

Use a statistical software or calculator to find the correlation between the Year and CO2 Concentrations. The result should range between -1 and 1.

Calculate Regression Line

Use a statistical software or calculator to calculate the regression line which will be in the form of \(y = MX + C\), where M is the slope of the line, and C is the intercept.

Interpret Slope

The slope of the regression line represents the rate of change in CO2 concentrations per unit increase in year. If the slope is positive, CO2 concentrations are increasing with time, and vice versa.

Understand Intercept

The intercept is the predicted CO2 concentration when the year is zero. As 'Year' in this case represents years starting from 1960, the intercept might not make sense contextually, depending on its value.

Predict CO2 Levels for 2003 and 2020

Substitute the values 2003 and 2020 for 'Year' in the regression equation. The resulting value for each is the estimated CO2 level for those years.

Find the Residual for 2010

The residual can be calculated by subtracting the predicted CO2 concentration for 2010 from the observed value. This quantifies the error in our prediction.

Key concepts

Explanatory Variable

The explanatory variable, also known as the independent variable, is the one that influences or explains changes in the response variable. Think of it as the cause in a cause-and-effect relationship. In the context of the carbon dioxide concentration problem, the explanatory variable is the 'Year'. This makes sense because as time progresses, we would expect some kind of trend or change in the levels of atmospheric CO₂, either due to natural patterns or human activity.

Knowing which is the explanatory variable is vital for regression analysis, as it helps determine the direction of the analysis—predicting the outcome ('response variable') based on the cause ('explanatory variable').

Response Variable

The response variable, also called the dependent variable, is the one affected by the explanatory variable. It's essentially the effect in the cause-and-effect relationship established by the regression analysis. In our example, 'CO₂ Concentration' is the response variable, since it is what we are trying to predict or explain as it changes over 'Year.'

Understanding the role of the response variable is important to correctly interpret the outcome of the analysis and to make meaningful predictions or inferences about future trends.

Scatterplot

A scatterplot is a type of graph that visually displays the relationship between two numerical variables. Each point on the scatterplot represents an individual data point. When plotting 'Year' on the x-axis and 'CO₂ level' on the y-axis, we can observe how the 'CO₂ Concentrations' vary over time. A scatterplot is fundamental in determining if there is a visual relationship—linear or nonlinear—between the variables.

A clear trend or pattern in the scatterplot can suggest a potential relationship worth investigating with further statistical tools like correlation coefficients and regression lines.

Correlation Coefficient

The correlation coefficient is a numerical measure of the strength and direction of a linear relationship between two variables. Its values range from -1 to 1, where:

• A value close to 1 indicates a strong positive linear relationship.
• A value close to -1 shows a strong negative linear relationship.
• A value close to 0 suggests little to no linear correlation.

Finding out the correlation coefficient between the 'Year' and 'CO₂ levels' helps us confirm if the visual interpretation from our scatterplot is statistically valid. A high correlation would support a linear model for prediction, as indicated in the example dataset.

Regression Line

The regression line, or the line of best fit, is a straight line that best represents the data on a scatterplot. It's the visual representation of the regression equation, which typically takes the form of \(y = mx + c\), where \(m\) is the slope, and \(c\) is the y-intercept. The purpose of this line is to predict the response variable based on the explanatory variable.

When analyzing the levels of CO₂ over the years, the regression line helps us understand the overall trend and make predictions about future CO₂ concentrations. The accuracy of these predictions is dependent on the strength of the linear relationship, as represented by the correlation coefficient.

Slope Interpretation

The slope of the regression line represents the average change in the response variable for every one-unit increase in the explanatory variable. In simpler terms, it shows how much 'Y' changes when 'X' increases by one. Interpreting the slope helps us understand the nature of the relationship between the variables.

In the exercise, the slope tells us the average increase in atmospheric CO₂ concentration for each year that passes. If the slope is positive, it means that CO₂ concentrations are generally increasing as the years progress.

Y-intercept Analysis

The y-intercept of the regression line is the point where the line crosses the y-axis. It reflects the predicted value of the response variable when the explanatory variable is zero. Y-intercept analysis can be tricky because it may not always make practical sense, depending on the context.

In the CO₂ concentration problem, the y-intercept would hypothetically represent the level of CO₂ in the year when 'Year' is zero. Since our 'Year' starts from 1960, the y-intercept can't be directly interpreted as the CO₂ level for the year 0. Instead, the intercept can help us understand the starting level of our data within the range of years examined.

Residual Calculation

The residual in regression analysis is the difference between the observed value of the response variable and the value predicted by the regression line. It reflects the error in the prediction for a specific data point. Residuals are important in diagnostics to assess the performance of the regression model.

Calculating a residual, as for the year 2010 in the provided example, involves taking the actual CO₂ concentration and subtracting the concentration predicted by our regression model for that same year. This specific residual tells us how off our model's prediction was for 2010, serving as a piece of the puzzle in evaluating the overall model accuracy.