Problem 39 - 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. $$ (1,4),(2,6),(3,8),(4,11),(5,13),(6,15) $$

Step-by-Step Solution

Answer

The given data has a positive linear correlation and depending on the calculated correlation coefficient using a graphing utility, it should be close to 1, indicating a strong positive relationship.

Step 1: Plotting the data points

The first step is to plot the given points on a graph. The points are (1,4),(2,6),(3,8),(4,11),(5,13),(6,15).

Step 2: Determine the type of correlation

Look at the plotted points. If the points seem to form a line going up to the right, then it's a positive linear correlation. If the points form a line going down to the right, then it's a negative linear correlation. If the points don't form any sort of line, then there is no linear correlation. In this case, as the x values increase, so do the y values which indicates a positive linear correlation.

Step 3: Calculate the correlation coefficient

To confirm the result, calculate the correlation coefficient (r value) using a graphing utility. This can be done by inputting the data points and applying the function for calculating the correlation coefficient.

Key Concepts

Linear Correlation

Linear correlation refers to a relationship between two variables, where they change together in a consistent pattern given by a straight line. To determine if a linear correlation exists:

• First, plot the data points on a graph.
• Observe the shape of the points. A positive linear correlation occurs when points form a line that slopes upward to the right. This means as one variable increases, the other one increases too.
• A negative linear correlation is present when points form a line that slopes downward to the right, indicating as one variable increases, the other one decreases.
• No correlation is observed when points are scattered without forming any clear pattern.

In our given example exercise, the data points such as (1,4), (2,6), and (3,8) suggest a positive linear correlation, as they rise alongside both the axes consistently.

Data Plotting

Data plotting is an essential step in exploring the relationship between variables. When you plot data, you visually represent the information, making it easier to see patterns and potential correlations. In most cases:

• Start by marking each data point on a graph. With coordinates like (1,4) or (2,6), the first number represents the x-axis and the second number represents the y-axis.
• Continue this process for all data points provided.

By marking the points on the grid, you get a sense of the overall distribution and can easily spot trends. With our dataset, plotting was key in recognizing the linear pattern indicating a positive correlation. This visual inspection is the first step before making quantitative calculations like the correlation coefficient.

Graphing Utilities

Graphing utilities are tools designed to help in the accurate calculation and visualization of mathematical concepts, like correlation coefficients. These tools often come in the form of software or online calculators that allow you to input data sets to perform automatic analyses.

• By using these utilities, you can calculate the correlation coefficient, denoted as \( r \), which quantifies the strength and direction of a linear relationship.
• An \( r \) value closer to 1 implies a strong positive correlation, while a value closer to -1 indicates a strong negative correlation.
• An \( r \) value near 0 would suggest little to no linear relationship between the data points.

For our exercise, a graphing utility confirmed the visible trend, providing an \( r \) value close to 1, thereby verifying the strong positive correlation seen in the plotted data.