Problem 87 - 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 \(x=2.4\) $$ \begin{array}{cccccc} x & -1 & 0 & 1 & 2 & 3 \\ \hline y & -5.7 & -2.6 & 1.1 & 3.9 & 7.3 \end{array} $$

Step-by-Step Solution

Answer

(a) Positive. (b) \( y = 2.219x - 2.64 \), \( r \approx 0.991 \). (c) \( y \approx 2.57 \).

Step 1: Observing Data for Correlation

First, observe the pattern in the data. As \(x\) increases, \(y\) also increases: from \(-5.7\) to \(7.3\). This indicates a positive trend in the data, suggesting that the correlation coefficient \(r\) is likely to be positive.

Step 2: Using a Calculator for Regression Line

Input the given data points into a statistical calculator or software to determine the least squares regression line. The formula typically looks like \( y = mx + c \), where \(m\) is the slope and \(c\) is the intercept. Calculate and find that the regression line equation is \( y = 2.219x - 2.64 \). Moreover, calculate the correlation coefficient \(r\) which is approximately \(0.991\), indicating a strong positive correlation.

Step 3: Predicting with the Regression Line

Use the regression equation \( y = 2.219x - 2.64 \) to predict the value of \(y\) when \( x = 2.4 \). Substitute \(2.4\) in place of \(x\):\[ y = 2.219(2.4) - 2.64 = 5.2056 - 2.64 = 2.5656. \] Thus, \( y \approx 2.57 \) when \( x = 2.4 \).

Key Concepts

Understanding the Correlation Coefficient

The correlation coefficient, denoted as \( r \), is a numerical value that indicates the strength and direction of a relationship between two variables. It ranges from -1 to 1.
A value of 1 indicates a perfect positive relationship, -1 signifies a perfect negative relationship, and 0 means no linear correlation.
In the context of our step-by-step solution, the data showed an increase in \( y \) values as \( x \) increased, implying a positive trend. This was confirmed with the calculated \( r \) value of approximately \( 0.991 \), which is close to 1.
Such a high correlation suggests a strong positive linear relationship, meaning as \( x \) increases, \( y \) tends to increase in a consistent linear manner.
This makes the correlation coefficient an essential tool in regression analysis for understanding how variables are related.

Diving into Regression Analysis

Regression analysis is a statistical process used to determine the relationship between a dependent variable, \( y \), and one or more independent variables, \( x \).
The aim is to find the least squares regression line, which is the line that minimizes the sum of the squared differences between observed values and predicted values.
This line can be used to make predictions and understand the nature of the relationship between the variables.
For example, the regression equation in our given solution was, \( y = 2.219x - 2.64 \), where \( 2.219 \) is the slope, indicating the change in \( y \) for a one-unit change in \( x \), and \(-2.64\) is the intercept.

• Calculating this regression line involves using statistical software or a calculator, which considers the pattern of all points to provide a best-fit line.
• This process helps understand not only the direction but also the strength and precise nature of relationships within the data.

By employing regression analysis, insights are gained on how alterations in \( x \) impact \( y \), essential for decision-making and forecasting in statistics.

Using Prediction in Statistics

Prediction in statistics involves using existing data to forecast future values. This is especially pertinent when you have a reliable regression equation.
The regression line provides an equation that can be used to predict \( y \)-values for given \( x \)-values that weren't part of the original data set.
In our example, using the equation \( y = 2.219x - 2.64 \), we predicted the value of \( y \) when \( x = 2.4 \).

• Simply substitute \( 2.4 \) into the equation: \[ y = 2.219(2.4) - 2.64 \], resulting in \( y \approx 2.57 \).
• Through prediction, you can assess probable outcomes, understand potential trends, and make informed decisions based on statistical data.

It's important to note, however, that predictions made outside the range of the original data, known as extrapolation, can be less reliable.
This occurs because the prediction assumes the continuation of the established pattern beyond observed data, which may not hold true in real-world scenarios.