"Old Vehicles In Exercises 31–34, use the figure shown at the left.

33. Coefficient of Determination Find the coefficient of determination r^2 and interpret the results."

Verified step by step guidance

1. Step 1: Identify the variables and data points. Here, the independent variable (x) is the year, and the dependent variable (y) is the average age of vehicles. The data points are given for years 2014 to 2021 with corresponding average ages.
2. Step 2: Calculate the correlation coefficient (r) between the year (x) and the average age (y). This involves finding the covariance of x and y, and dividing it by the product of their standard deviations. The formula is: $r=\frac{\mathrm{cov(x,y)}}{\sqrt{\mathrm{var(x)}\cdot \mathrm{var(y)}}}$.
3. Step 3: Once you have the correlation coefficient r, calculate the coefficient of determination r² by squaring r. This value represents the proportion of the variance in the dependent variable (average age) that is predictable from the independent variable (year).
4. Step 4: Interpret the coefficient of determination r². A value close to 1 indicates a strong linear relationship where the year explains most of the variation in average vehicle age. A value close to 0 indicates a weak relationship.
5. Step 5: Summarize your findings by stating how well the year predicts the average age of vehicles based on the calculated r² value.

Key Concepts

Coefficient of Determination (r²)

The coefficient of determination, denoted as r², measures the proportion of the variance in the dependent variable that is predictable from the independent variable. It ranges from 0 to 1, where a higher value indicates a better fit of the regression model to the data. In this context, it shows how well the year explains changes in the average vehicle age.

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation. Here, it helps to analyze how the average age of vehicles changes over the years, providing a basis to calculate r² and interpret trends.

Interpretation of Statistical Results

Interpreting statistical results involves understanding what the calculated values imply in real-world terms. For r², this means explaining how much of the variation in vehicle age is explained by the year, which helps in assessing trends and making informed conclusions about vehicle longevity over time.