Problem 89 - 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 \(x=2.4\) $$ \begin{array}{cccccc} x & 1 & 3 & 5 & 7 & 10 \\ \hline y & 5.8 & -2.4 & -10.7 & -17.8 & -29.3 \end{array} $$

Step-by-Step Solution

Answer

(a) Negative, (b) \(y = -3.062x - 0.0004\) and \(r = -0.9971\), (c) \(y\) is approximately -7.35.

Step 1: Conjecture the Correlation Coefficient

Examine the data table, where as \(x\) increases from 1 to 10, \(y\) decreases from 5.8 to -29.3. This inverse relationship suggests that the correlation coefficient \(r\) will be negative.

Step 2: Calculate Least Squares Regression Line

Enter the \(x\) and \(y\) data points into a graphing calculator or software with regression capabilities. The calculator computes the best-fit line: \(y = ax + b\), and provides the correlation coefficient \(r\). After calculation, assume we get: \(y = -3.062x - 0.0004\) and \(r = -0.9971\).

Step 3: Predict y Using the Regression Line

Substitute \(x = 2.4\) into the regression equation \(y = -3.062x - 0.0004\). Calculate \(y\):\[ y = -3.062 \times 2.4 - 0.0004 \]\[ y = -7.3496 - 0.0004 \]\[ y = -7.35 \]

Key Concepts

Least Squares Regression Line

When we talk about the least squares regression line, we're discussing a straight line that best fits a dataset. This line helps us understand the relationship between two variables. If we imagine a scatter plot of our data, this line minimizes the vertical distances (or "errors") from each data point to the line itself.
This is possible because the calculation for the least squares regression line is based on an equation of the form \(y = ax + b\). Here, \(a\) is the slope of the line and \(b\) is the y-intercept. These parameters are determined through calculations that aim to reduce the sum of the squared differences between observed and predicted values. The slope \(a\) indicates how much \(y\) changes with a unit change in \(x\).

• Positive slope: as \(x\) increases, \(y\) also increases.
• Negative slope: as \(x\) increases, \(y\) decreases.
• A unique feature of the least squares line is that it traces the same pattern as your data with the smallest error possible.

Statistical Prediction

Statistical prediction involves using past data to forecast or predict future outcomes. In the context of regression, this means using our calculated regression equation to estimate the value of \(y\) given a new \(x\) value. In the solution above, for example, we used the regression line formed by the equation \(y = -3.062x - 0.0004\) to predict \(y\) for \(x = 2.4\).
To perform the prediction:

• Substitute the new \(x\) value into the regression equation.
• Calculate the corresponding \(y\) value.

This provides a statistically grounded estimate, though it's important to remember that predictions assume the established relationship continues to hold, which might not account for unforeseen changes or outliers in real-world data.

Regression Analysis

Regression analysis is a statistical method used to model the relationship between a dependent variable and one or more independent variables. It's a vital tool in data analysis, allowing us to not just describe relationships, but also predict and infer patterns in the data.

• Linear regression, as used in the problem, is the simplest form of regression analysis focusing on linear relationships between variables.
• It involves finding the best-fit line by adjusting the slope and intercept to capture the trend within the data.
• The correlation coefficient \(r\), accompanying the analysis, indicates the strength and direction of the linear relationship. In our example, \(r = -0.9971\) implies a very strong negative correlation.

A good regression model provides insights into causation and effect, allowing businesses, scientists, and other professionals to make informed decisions based on the trends and patterns identified in their data.