Problem 42 - 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 $$ (0.5,2),(0.75,1.75),(1,3),(1.5,3.2),(2,3.7),(2.6,4) $$

Step-by-Step Solution

Answer

First plot the given points on a graph. Based on the graph, make a visual interpretation if the data points have a positive, negative, or no correlation. Then, calculate the correlation coefficient (\(r\)) using a graphing tool. This numerical value should confirm your visual interpretation.

Step 1: Plotting the points

Begin by plotting the provided points on a graph. The points are: (0.5,2), (0.75,1.75), (1,3), (1.5,3.2), (2,3.7), (2.6,4).

Step 2: Visual interpretation of the graph

Once the points are plotted, visually examine the graph to determine if there is any pattern or trend. If the plotted points appear to rise from left to right, then they indicate a positive correlation. If they seem to fall from left to right, it suggests a negative correlation. If no clear pattern or trend emerges, then there is no correlation.

Step 3: Calculation of the correlation coefficient (\(r\))

To confirm the visual interpretation, calculate the correlation coefficient (\(r\)) using a graphing utility. This will provide a quantitative measure of the correlation.

Step 4: Confirming the result

Finally, compare the calculated \(r\) with the visual interpretation from the graph. If \(r\) is close to 1, it means there is a strong positive correlation. If it's close to -1, there is a strong negative correlation. An \(r\) around 0 indicates no correlation. These should correspond with what you observed from your graph.

Key Concepts

Linear Correlation

Linear correlation is a technique used to describe the relationship between two variables. When we say there is a "linear" correlation, we mean that the relationship between these variables can be represented by a straight line when plotted on a graph. This concept is fundamental to statistics and helps in understanding whether changes in one variable directly affect the other.
Understanding linear correlation involves computing the correlation coefficient, denoted as \( r \). This value ranges from -1 to 1 and quantifies the degree to which two variables are linearly related. The closer the value of \( |r| \) is to 1, the stronger the linear relationship.
When plotting data, if the points align well with a line through the graph, this indicates a linear correlation. Recognizing these patterns visually can be the first step before further mathematical analysis.

Plotting Data Points

Plotting data points is an essential step in analyzing data sets. It involves marking each pair of values on a coordinate plane, where each point represents two related numbers from the data set.
To plot the data points:

• Start by drawing a set of axes: a horizontal axis (usually representing one variable) and a vertical axis (for the other variable).
• Mark each pair of values as a point. For example, the point (0.5, 2) means you go 0.5 units on the x-axis and 2 units up on the y-axis.

Once all the points are plotted, you can begin to look for patterns.
Do the points seem to form a line? Or, do they scatter around randomly?
Plotting is not only the first step in correlation analysis but also a helpful visual tool to predict trends and relationships between the variables.

Positive Correlation

Positive correlation occurs when two variables have a relationship where one variable increases when the other one does. This is visually represented in a graph where all data points tend to rise from left to right.
For example, consider the set of points given: (0.5,2), (0.75,1.75), (1,3), (1.5,3.2), (2,3.7), (2.6,4). When these are plotted, if they form an upward-sloping line, this indicates positive correlation.

• The higher the \( |r| \) value is towards 1, the stronger the positive linear relationship.
• A strong positive correlation means new data points will likely fall near this line.

Positive correlation is often used in scenarios where growth or increase in one variable leads to similar increase in another, like study time and test scores. This predictive power of correlation makes it a vital tool in data analysis.

Negative Correlation

Negative correlation describes a relationship between two variables where one variable decreases as the other increases. When you plot your data points on a graph, a negative correlation is indicated by a downward slope from left to right.
For example, if we had a different set of points that trended downward, this would suggest a negative correlation. In such cases:

• The correlation coefficient \( r \) would be close to -1, indicating a strong negative linear relationship.
• Negative correlation is utilized to analyze situations where one action or set of events causes a decrease in another variable.

Understanding negative correlation is critical in fields like finance, where an inverse relationship might help predict trends, such as an increase in interest rates leading to a decrease in investment. Recognizing these trends visually and mathematically helps make informed decisions based on data.