Problem 85 - College Algebra with Modeling and Visualization

Question

Exercises 85 and 86: Find the line of least-squares fit for the given data points. What is the correlation coefficient? Plot the data and graph the line. $$ (-2,2),(1,0),(3,-2) $$

Step-by-Step Solution

Answer

Line: \(y = -0.474x + 0.317\); Correlation Coefficient: \(-0.48\).

Step 1: Calculate the Means

To find the line of least-squares fit, we first calculate the means of the x-values and y-values. Data points: \((-2, 2), (1, 0), (3, -2)\)Mean of x-values (\( \bar{x} \)) = \( \frac{-2 + 1 + 3}{3} = \frac{2}{3} \approx 0.67\)Mean of y-values (\( \bar{y} \)) = \( \frac{2 + 0 - 2}{3} = 0\).

Step 2: Compute the Components of the Slope Formula

Next, we calculate the components of the formula for the slope of the least-squares regression line. This involves finding certain sums.The formula for slope \(m\) is: \[ m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}} \]Compute the components:\( \sum{(x_i - \bar{x})(y_i - \bar{y})} = (-2 - 0.67)(2 - 0) + (1 - 0.67)(0 - 0) + (3 - 0.67)(-2 - 0) = -1.34 + 0 - 4.66 = -6\)\( \sum{(x_i - \bar{x})^2} = (-2 - 0.67)^2 + (1 - 0.67)^2 + (3 - 0.67)^2 = 7.11 + 0.11 + 5.44 = 12.66\).

Step 3: Calculate the Slope of the Line

Using the components computed, we can find the slope \(m\):\[ m = \frac{-6}{12.66} \approx -0.474 \]

Step 4: Calculate the Y-intercept of the Line

With the slope \(m\) known, calculate the y-intercept \(b\) using the point-slope form of the line's equation:\[ b = \bar{y} - m\bar{x} = 0 - (-0.474 \times 0.67) \approx 0.317 \]Thus, the equation of the line of best fit is:\[ y = -0.474x + 0.317 \]

Step 5: Calculate the Correlation Coefficient

The correlation coefficient \( r \) quantifies the strength of the linear relationship between the two variables. It can be calculated as:\[ r = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\sum{(x_i - \bar{x})^2} \cdot \sum{(y_i - \bar{y})^2}}} \]Where:\( \sum{(y_i - \bar{y})^2} = (2-0)^2 + (0-0)^2 + (-2-0)^2 = 8\)\[ r = \frac{-6}{\sqrt{12.66 \times 8}} \approx -0.48 \]

Step 6: Plot the Data and Line of Best Fit

Plot the points (-2, 2), (1, 0), and (3, -2) on a coordinate plane. Draw the line represented by the equation \( y = -0.474x + 0.317 \). Ensure the line fits closely to the data points, illustrating the direction and strength of the association between x and y.

Key Concepts

Correlation Coefficient

The correlation coefficient, denoted as \( r \), is a statistical measure that illustrates how strongly two variables are related. In the context of least-squares regression, it helps us understand the degree and direction of the linear relationship between the variables involved. For instance, in our original exercise, it is calculated by using a specific formula that takes into account the deviations of data points from their respective means.

The correlation coefficient value ranges from \(-1\) to \(1\). A value of \(1\) indicates a perfect positive linear relationship, while \(-1\) signifies a perfect negative linear relationship. A value of \(0\) implies no linear relationship. In this exercise, we found \( r \approx -0.48 \). This negative value suggests a moderate inverse relationship between the x and y variables. It indicates that as the x-values increase, the y-values tend to decrease.

Slope Calculation

Calculating the slope of the line of best fit is an important step in the least-squares regression process. It involves determining the rate of change between the dependent and independent variables.

In this exercise, the slope \( m \) is found using the formula:

• \( m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}} \)

This formula basically computes the average of the changes in y relative to x, adjusting for their deviations from their means. Once the sums were calculated, we found that \( m \approx -0.474 \).

This negative slope indicates that there’s a decreasing trend in the relationship between the two variables given in the exercise. As x increases by 1 unit, y decreases by approximately 0.474 units.

Line of Best Fit

The line of best fit, also known as the regression line, represents the best possible linear relationship that minimizes the discrepancies between the actual data points and the points predicted by the line.

Using the least-squares method ensures that the sum of the squares of the vertical distances (errors) from the data points to the line is minimized. This is why it is sometimes called the "least squares" fit.

• The equation of the line of best fit is generally given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

In the solution to the exercise, after calculating both the slope and the y-intercept, the line of best fit was determined to be \( y = -0.474x + 0.317 \). This line ties together the data and gives us a visual representation of how the data behaves.

Scatter Plot

A scatter plot is a type of graph used to visually display and assess the values of two different variables. It helps in understanding the relationships, patterns, or trends between them.

In this exercise, a scatter plot was created by plotting the provided data points:

• (-2, 2)
• (1, 0)
• (3, -2)

Once the data points were plotted, the line of best fit \( y = -0.474x + 0.317 \) was drawn to see how closely it conformed to the trend suggested by the scatter of the data points.

This visual representation aids in confirming the correlation coefficient's suggestion of a moderate negative association between x and y. Scatter plots are fundamental for initial data analysis as they provide a clear picture of possible correlations.