Problem 41 - College Algebra and 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 coefficient. 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. $$ (1,3),(2,6),(3,2),(4,3),(5,9),(6,1) $$

Step-by-Step Solution

Answer

The answer depends on the observation from the graph and calculated correlation coefficient \(r\).

Step 1: Plot the points

On a suitable scale and set of axes, plot the points (1,3), (2,6), (3,2), (4,3), (5,9), (6,1). The horizontal axis represents the x-values and the vertical axis represents the y-values of the given points.

Step 2: Determine the type of correlation

Carefully observe the plotted points. If they appear to 'rise' from left to right, the correlation is positive. If they 'fall' from left to right, the correlation is negative. If the points does not appear to follow a general trend, they do not have a linear correlation.

Step 3: Calculate the correlation coefficient (\(r\))

Enter the pairs of data points in a graphing calculator or any online tool to calculate the correlation coefficient \(r\). This value will confirm the type of linear relationship initially visually identified.

Key Concepts

Linear Correlation

Linear correlation is a statistical measure that indicates the extent to which two variables fluctuate together. When we plot a set of points on a graph, the pattern of those points can suggest different kinds of relationships.

For instance, if the points generally tend to move upward from left to right, this suggests a positive linear correlation. It means that as one variable increases, so does the other. Conversely, a downward trend indicates a negative linear correlation, signifying that as one variable increases, the other decreases.

If the plotted points are widely scattered with no apparent trend, they exhibit no linear correlation. The correlation coefficient, denoted as \(r\), quantifies this relationship. Values of \(r\) range from -1 to 1, with 1 implying a perfect positive linear correlation, -1 a perfect negative linear correlation, and 0 no correlation at all. The magnitude of \(r\) gives us an idea about the strength of the correlation; the closer \(|r|\) is to 1, the stronger the correlation.

Graphing Utility

A graphing utility is an essential tool for anyone dealing with data. It is a software or graphing calculator that allows users to plot data points and visualize the relationship between them visually.

Graphing utilities offer a convenient way to calculate statistical information, such as correlation coefficients. They are equipped with functionalities to perform regression analysis, which helps to understand how variables relate to each other and can be used to predict future trends.

Utilizing Graphing Utilities

With a graphing utility, you can input your data points, such as \((1,3),(2,6),(3,2),(4,3),(5,9),(6,1)\), and the software immediately plots these points on a coordinate plane for you. Sophisticated models even provide curve-fitting functionality, which draws a line that best fits the plotted data. This is critical when requiring a visual representation of the data's trend, and such tools often come with the option to calculate the correlation coefficient directly.

Data Plotting

Data plotting is the process of graphically representing data points on a coordinate plane. It serves as a visual tool that aids in the understanding of how the data points relate to each other.

When plotting data, we decide on a scale that best represents our data sets on axes, typically X(horizontal axis) for independent variable and Y(vertical axis) for dependent variable. The points you get from pairing these values - for example, from the dataset \((1,3),(2,6),(3,2),(4,3),(5,9),(6,1)\), form a scatter plot.

The Significance of Data Plotting

By visually assessing a scatter plot, we can initially infer the possible relationship between variables. It's the first step in identifying the type of correlation before further computational methods are applied. It’s also a foundational skill for students as they progress to more complex concepts involving data analysis. Well-plotted data can reveal trends, clusters, outliers, or patterns that may not be immediately obvious through numerical analysis alone.