Problem 44 - 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. $$ (0.5,9),(1,8.5),(1.5,7),(2,5.5),(2.5,5),(3,3.5) $$

Step-by-Step Solution

Answer

The data points show a negative correlation and the correlation coefficient \(r\) calculated using the graphing utility should be close to -1.

Step 1: Plot the points

Plot the following points on a graph: (0.5,9), (1,8.5), (1.5,7), (2,5.5), (2.5,5), (3,3.5). Do it accurately and use a graph paper for best results. To plot a point, its first number is its horizontal position (the x-coordinate), and the second number is the vertical position (the y-coordinate).

Step 2: Assess the correlation

Look at the graph. If the points are ascending to the right, it suggests a positive correlation. If they're descending to the right, it would be a negative correlation. If there's no apparent pattern, there might be no correlation. In this case, the points descend to the right, so the correlation is likely negative.

Step 3: Use a graphing utility to find \(r\)

With the points plotted on the graph, use a graphing utility to calculate the correlation coefficient \(r\). Many scientific graphing calculators and software have the capability to calculate the correlation coefficient. In each case, the actual procedure can vary slightly, so refer to the particular user manual or help files. In general, you need to enter the pairs of data points, then you'll find \(r\) in the statistical analysis menu or functions.

Step 4: Confirm your result

The value of \(r\) calculated by the graphing utility should confirm your assessment from Step 2. If \(r\) is close to -1, then indeed there's a negative correlation, which is what we expected because of the graphical examination. If that's the case, then your initial assessment was correct.

Key Concepts

Understanding Linear Correlation

Linear correlation is a statistical measure that describes the strength and direction of a relationship between two variables. When data points on a graph can be connected to form a line, they are said to have a linear correlation. This correlation can be positive, negative, or non-existent (no correlation).

A positive linear correlation means that as one variable increases, the other variable tends to increase as well. Conversely, a negative linear correlation indicates that as one variable increases, the other tends to decrease. No correlation suggests there is no discernible pattern in the increase or decrease of the variables when compared to each other.

In the example exercise, when plotting the points on a graph, a pattern emerges that suggests a negative linear correlation. As the x-values increase, the y-values decrease, which is characteristic of a negative relationship. Understanding this concept is fundamental to identifying patterns within data sets and can have numerous applications across different fields of study.

Utilizing Graphing Utilities

A graphing utility is a tool that can plot graphs, manage statistical data, and perform various types of analysis. These can include scientific graphing calculators, computer software, or online platforms specifically designed to handle complex mathematical functions.

To accurately determine the correlation coefficient, otherwise known as 'r', graphing utilities provide immense help. They can process the given data points and output the value of 'r' with high precision. As mentioned in the solution steps, after plotting the points, the graphing utility can be used to calculate 'r', which tells us how well a straight line can describe the relationship between the data points.

How to Use a Graphing Utility:

• Input the pairs of data points into the utility.
• Navigate the features to find the 'statistical analysis' function.
• Execute the function to find the correlation coefficient 'r'.

This step streamlines the process, making it not only quicker but also more accurate than manual calculation.

The Importance of Data Visualization

Data visualization is the representation of data in a visual context, like a chart or graph, to help people understand the significance of data by placing it in a visual context. Patterns, trends, and correlations that might go undetected in text-based data can be exposed and recognized more easily with data visualization.

In our textbook exercise, data visualization comes into play when we plot the data points. The visual representation helps in quicker identification of the type of correlation between variables. This visual assessment is crucial as it gives an initial hypothesis before any calculations are done.

Why Data Visualization Matters:

• Helps to quickly convey messages.
• Makes complex data more accessible.
• Identifies areas that require attention or improvement.
• Clarifies which factors influence customer behavior.
• Predicts sales volumes.

Effective data visualization can greatly enhance the understanding and communication of data insights.

Statistical Analysis in Determining Correlation

Statistical analysis involves collecting and scrutinizing every data sample in a set of items from which samples can be drawn. A sample, in this context, is a set of data collected and/or selected from a statistical population by a defined procedure. The analysis allows us to make interpretations and extract conclusions based on data patterns.

In the context of the exercise, after visualizing the data through graphing, further statistical analysis is done by computing the correlation coefficient ('r'). This coefficient quantifies the degree of linear relationship between the variables.

Key Points in Statistical Analysis:

• It facilitates the understanding of data.
• It helps in making informed decisions based on data.
• Identifying the correlation coefficient is part of descriptive statistics.
• It applies to various fields such as economics, biology, engineering, etc.

The computed 'r' value can either confirm or refute our initial visual assessment and is essential for evidence-based decision-making.