Problem 44 - Calculus an Applied Approach

Question

Plot the points and determine whether the data have positive, negative, or no linear correlation (see figures below). Then use a graphing utility to find the value of \(r\) and confirm your result. The number \(r\) is called the correlation confficient. It is a measure of how well the model fits the data. Correlation coefficients vary between \(-1\) and \(1,\) and the closer \(|r|\) is to \(1,\) the better the model. (GRAPH NOT COPY) Positive correlation Negative correlation No correlation $$ (0.5,9),(1,8.5),(1.5,7),(2,5.5),(2.5,5),(3,3.5) $$

Step-by-Step Solution

Answer

The data points (0.5,9), (1,8.5), (1.5,7), (2,5.5), (2.5,5), (3,3.5) have a negative linear correlation, as observed from the scatterplot. The correlation coefficient, \(r\), can be calculated using a graphing tool to confirm this.

Step 1 - Plot the points

For the set of data provided, start by plotting the points on a graph. They are (0.5,9), (1,8.5), (1.5,7), (2,5.5), (2.5,5), (3,3.5).

Step 2 - Analyze the plot

Look at the plotted points. If there's a pattern in which the points ascend or descend in an almost straight line, there's a correlation. Since the points descend, there is a negative correlation.

Step 3 - Find the correlation coefficient

With a graphing utility (like a calculator or a computer software), input your data points to calculate the value of \(r\). The value should be between -1 and 1. A value close to -1 indicates a strong negative linear correlation.

Step 4 - Confirm your result

Verify if the correlation coefficient \(r\) you find matches with your prior analysis whether the data have positive, negative, or no correlation.

Key Concepts

Linear Correlation

When we speak of linear correlation, we refer to the relationship between two variables where if one variable increases, the other variable tends to also increase (positive correlation) or decrease (negative correlation) in a consistent pattern. If the pattern forms a straight line when we plot the variables against each other on a graph, we say they have a strong linear correlation. The strength of this relationship is quantified by the correlation coefficient, denoted as r.

The concept of linear correlation is at the heart of many statistical analyses, providing insights into how different quantities relate to and influence each other. For example, height and shoe size often display a positive linear correlation; as one's height increases, their shoe size tends to increase as well.

Plotting Data Points

To better understand the relationship between two variables, we begin by plotting data points on a graph, typically using a Cartesian coordinate system. Each data point represents a pair of values, with the first value indicating the position on the x-axis (horizontal axis) and the second value on the y-axis (vertical axis). By visually assessing the plot, we can initially determine if a linear relationship exists.

If the data points form a discernible line, ascending or descending as we move along the x-axis, this suggests a linear correlation, positive or negative respectively. If there is no apparent line, and the points are scattered without any direction, it suggests no correlation—this indicates that the two variables do not have a linear relationship.

Graphing Utility

A graphing utility is an indispensable tool for statisticians, mathematicians, and anyone working with data. It could be a graphing calculator, computer software, or an online tool that assists in plotting data points, performing various calculations, and analyzing relationships between variables.

These utilities help users visualize the trend of the data points and usually have built-in functions to calculate the correlation coefficient r. Accurate calculation of r would be complex by hand but is efficiently managed by these tools, thus confirming the type of correlation and its strength quickly, which allows for more time to interpret and analyze the data.

Correlation Analysis

Correlation analysis involves determining the degree to which two variables are related. The correlation coefficient r is the numerical measure that expresses the strength and the direction of a linear relationship between two variables. Values of r close to 1 indicate a strong positive correlation, whereas values close to -1 indicate a strong negative correlation. An r near 0 suggests no evident linear relationship.

It's critical to note that correlation does not imply causation. Even if two variables display a strong correlation, it does not mean that one variable directly causes the changes in the other. There may be other underlying factors or variables at play, and further analysis would be needed to understand the causal relationships, if any.