Is a linear model appropriate for the relationship between these two variables? Discuss the scatterplot, correlation coefficient and residual plot to answer this question.

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Is a linear model appropriate for the relationship between these two variables? Discuss the scatterplot, correlation coefficient and residual plot to answer this question. 

### Analysis of Winning Times Over the Years

#### Scatterplot

The scatterplot illustrates the trend in winning times from 1950 to 2010. The x-axis represents the year, ranging from 1950 to 2010, while the y-axis indicates the winning times, measured in seconds, spanning from 27 to 29.6 seconds. Each blue dot signifies data for a particular year. The trendline shows a general decline in winning times over the years, suggesting an improvement in performance.

#### Residual Plot

The residual plot below the scatterplot displays the residuals of the winning times against the year. The x-axis represents the year, similar to the scatterplot, while the y-axis indicates the residual values, ranging from -0.8 to 0.8. The distribution of residuals helps assess the fit of the linear regression model. Most points lie near the zero line, suggesting a decent fit, though there are some deviations.

#### Regression Equation and Statistics

- **Regression Equation:** \( y = 93.7342 - 0.033x \)
- **Correlation Coefficient (r):** -0.843
- **Coefficient of Determination (r²):** 0.71

The negative correlation coefficient of -0.843 indicates a strong inverse relationship between the year and winning time. The \( r^2 \) value of 0.71 signifies that approximately 71% of the variability in winning times can be explained by the year.
Transcribed Image Text:### Analysis of Winning Times Over the Years #### Scatterplot The scatterplot illustrates the trend in winning times from 1950 to 2010. The x-axis represents the year, ranging from 1950 to 2010, while the y-axis indicates the winning times, measured in seconds, spanning from 27 to 29.6 seconds. Each blue dot signifies data for a particular year. The trendline shows a general decline in winning times over the years, suggesting an improvement in performance. #### Residual Plot The residual plot below the scatterplot displays the residuals of the winning times against the year. The x-axis represents the year, similar to the scatterplot, while the y-axis indicates the residual values, ranging from -0.8 to 0.8. The distribution of residuals helps assess the fit of the linear regression model. Most points lie near the zero line, suggesting a decent fit, though there are some deviations. #### Regression Equation and Statistics - **Regression Equation:** \( y = 93.7342 - 0.033x \) - **Correlation Coefficient (r):** -0.843 - **Coefficient of Determination (r²):** 0.71 The negative correlation coefficient of -0.843 indicates a strong inverse relationship between the year and winning time. The \( r^2 \) value of 0.71 signifies that approximately 71% of the variability in winning times can be explained by the year.
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