Problem 28 - College Algebra

Question

For the following exercises, use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy. $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline{x} & {4} & {5} & {6} & {7} & {8} & {9} & {10} & {11} & {12} & {13} \\ \hline y & {44.8} & {43.1} & {38.8} & {39} & {38} & {32.7} & {30.1} & {29.3} & {27} & {25.8}\\\ \hline\end{array}$$

Step-by-Step Solution

Answer

Regression line: \( y = -1.7x + 49.1 \), correlation coefficient: \( r = -0.971 \).

Step 1: Input Data

First, input the given data into your calculator or software. For this set: - X values: 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 - Y values: 44.8, 43.1, 38.8, 39, 38, 32.7, 30.1, 29.3, 27, 25.8. Ensure that each x-value is paired correctly with its corresponding y-value.

Step 2: Calculate the Regression Line

Use the regression function of your calculator or software to calculate the equation of the regression line. Generally, the calculator will output an equation in the form of:\[ y = mx + b \]where \( m \) is the slope and \( b \) is the y-intercept.

Step 3: Determine the Slope and Intercept

Once processed, you should find:- Slope \( m \approx -1.7 \)- Intercept \( b \approx 49.1 \)The regression equation is approximately \( y = -1.7x + 49.1 \).

Step 4: Calculate the Correlation Coefficient

The correlation coefficient \( r \) indicates the strength and direction of a linear relationship between x and y. Use your calculator's statistical function to find \( r \). Ensure to round this value to three decimal places.

Step 5: Record the Correlation Coefficient

After calculation, the correlation coefficient is found to be \( r \approx -0.971 \). This value suggests a strong negative correlation, meaning as \( x \) increases, \( y \) tends to decrease.

Key Concepts

Correlation Coefficient

The correlation coefficient, often represented by the symbol \( r \), is a key statistic for understanding the relationship between two variables. It ranges from -1 to 1. The value of \( r \) provides insights into how closely the two sets of data are correlated.

• If \( r = 1 \), there is a perfect positive linear relationship: as one variable increases, the other increases proportionally.
• If \( r = -1 \), there is a perfect negative linear relationship: as one variable increases, the other decreases proportionally.
• If \( r = 0 \), there is no linear relationship between the variables.

Values closer to 1 or -1 indicate stronger correlations. In our example, the correlation coefficient is approximately \( r = -0.971 \). This means there is a strong negative correlation between the variables, suggesting that as the \( x \) values increase, the \( y \) values tend to decrease sharply.

Linear Relationship

When a dataset exhibits a linear relationship, the data points lie closely along a straight line. This implies that one variable can be expressed as a linear function of another variable.

• This relationship can be depicted through the regression line.
• The strength and direction of this relationship are measured using the correlation coefficient.

For instance, in our case, the strong negative correlation coefficient \( r = -0.971 \) signifies that the data points align well with a downward-sloping line. Such linear relationships are substantial in predictive analytics, allowing for the estimation of one variable based on another.

Regression Line

A regression line is a straight line that best fits the data points on a scatter plot. This line is used to depict the relationship between two sets of data, providing the basis for predicting the dependent variable based on the independent variable. The equation for the best-fit line is typically given in the linear form:
\[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept.
The regression line obtained from our dataset is approximately \( y = -1.7x + 49.1 \). This means that, on average, for each one unit increase in \( x \), \( y \) decreases by 1.7 units, starting from a y-intercept of 49.1. Calculating the regression line helps in making predictions and understanding the extent to which changes in \( x \) affect \( y \).

Slope and Intercept

The slope and intercept are critical components of the regression line equation. They determine the orientation and position of the line.

• The slope, represented by \( m \), quantifies the steepness and the direction of the line. A positive \( m \) indicates an upward slope, while a negative \( m \) indicates a downward slope.
• The y-intercept, denoted by \( b \), is the point at which the line crosses the y-axis. This value represents the estimated value of \( y \) when \( x = 0 \).

In our analysis, the slope \( m \approx -1.7 \), and the intercept \( b \approx 49.1 \). These values suggest that as \( x \) increases, \( y \) decreases due to the negative slope. Understanding the slope and intercept is crucial for formulating hypotheses and comprehending the overall trend depicted by the data.