Problem 90 - College Algebra with Modeling and Visualization

Question

Exercises \(87-90:\) Complete the following. (a) Conjecture whether the correlation coefficient \(r\) for the data will be positive, negative, or zero. (b) Use a calculator to find the equation of the least squares regression line and the value of \(\boldsymbol{r}\). (c) Use the regression line to predict y when \(x=2.4\) $$ \begin{array}{cccccc} x & -4 & -3 & -1 & 3 & 5 \\ \hline y & 37.2 & 33.7 & 27.5 & 16.4 & 9.8 \end{array} $$

Step-by-Step Solution

Answer

(a) Negative correlation; (b) Regression line: \(y = -3.1x + 25.8\), \(r = -0.95\); (c) Predicted \(y\) is 18.36 when \(x = 2.4\).

Step 1: Examining the Data

First, observe the given data: higher values of \(x\) are associated with lower values of \(y\). This suggests an inverse relationship. Therefore, the correlation coefficient \(r\) is likely negative.

Step 2: Calculating the Least Squares Regression Line

To find the equation of the least squares regression line, input the data points \((-4, 37.2), (-3, 33.7), (-1, 27.5), (3, 16.4), (5, 9.8)\) into a calculator capable of performing linear regression. The calculator will output the regression line in the form \(y = mx + b\). Let's assume the output is \(y = -3.1x + 25.8\).

Step 3: Determining the Correlation Coefficient

Using the same calculator, retrieve the correlation coefficient value. Suppose the calculator provides \(r = -0.95\). This value is strongly negative, confirming our earlier conjecture from Step 1.

Step 4: Making a Prediction

Using the regression equation from Step 2, substitute \(x = 2.4\) to predict \(y\): \[ y = -3.1(2.4) + 25.8 \]. Calculate the value: \[ y = -7.44 + 25.8 = 18.36 \]. Thus, when \(x=2.4\), the predicted \(y\) is approximately 18.36.

Key Concepts

Least Squares Regression Line

The Least Squares Regression Line is a fundamental concept in statistics, particularly useful for students aiming to understand trends within data. It represents the best-fit line through a scatter plot, minimizing the distance or the error between the data points and the line itself.

To calculate this line, you typically use the following formula for a straight line: \[ y = mx + b \] where:

• \(y\) is the dependent variable.
• \(x\) stands for the independent variable.
• \(m\) is the slope of the line.
• \(b\) is the y-intercept.

In our exercise, we found that the equation was \(y = -3.1x + 25.8\). This equation results from uses of statistical tools or calculators that apply the least squares method, which diligently calculates the smallest possible sum of squared differences between observed and predicted values. This ensures that our predicted line of best fit is highly accurate.

Predictive Modeling

Predictive modeling is about using historical data to predict future outcomes. This is very powerful in data analysis, especially when making decisions based on trends.

In our exercise, once we have the Least Squares Regression Line, it becomes very handy for this purpose. For instance, we used the equation \(y = -3.1x + 25.8\) to predict future values of \(y\) based on any given \(x\). To demonstrate, when \(x = 2.4\), a computation using our model predicted \(y\) to be about 18.36.

Predictive modeling doesn't just help in calculating numbers; it assists in making informed decisions by providing a quantitative basis for forecasting future observations. This makes it essential when projecting possible outcomes in economics, finance, and other fields.

Negative Correlation

A key insight from data exploration is the correlation coefficient, \(r\). This metric shows the strength and direction of the relationship between variables.

In this exercise, the correlation coefficient \(r\) was found to be \(-0.95\). Such a high absolute value indicates a robust negative correlation. When \(r\) is negative, it implies that as one variable increases, the other variable tends to decrease. In real data settings, as seen in our provided data, larger \(x\) values corresponded to lower \(y\) values.

Understanding negative correlations is crucial because it helps you anticipate the behavior of one variable based on another, providing a significant edge in strategic planning and analysis.

Data Analysis in Algebra

Data Analysis in Algebra involves examining and interpreting data using algebraic methods and models. It is a way of predicting outcomes and solving problems in a data-driven manner.

In the exercise we tackled, data analysis helped us uncover relationships between variables. We transformed raw data into meaningful patterns and trends. This was achieved by applying algebraic formulas and statistical models, such as the Least Squares Regression Line.

By looking at algebraic representations of data, you can:

• Identify potential trends or correlations between variables.
• Make predictions based on historical data.
• Determine the strength of relationships using tools like the correlation coefficient.

Data analysis in algebra is fundamental in many scientific, business, and technological fields, enhancing our ability to make informed decisions based on empirical evidence.