The table gives the winning heights for the men's Olympicpole vault competitions up to the year $2004.$$\begin{array}{l}\text{ (a) Make a scatter plot and decide whether a linear model is }\\ \text{ appropriate. }\\ \text{ (b) Find and graph the regression line. }\\ \text{ (c) Use the linear model to predict the height of the winning }\\ \text{ pole vault at the }2008\text{ Olympics and compare with the }\\ \text{ actual winning height of }5.96\text{ meters. }\end{array}$$\begin{array}{l}\text{ (d) Is it reasonable to use the model to predict the winning }\\ \text{ height at the }2100\text{ Olympics? }\end{array}$

Key Concepts: Scatter Plot, Linear Model, Regression Line, Prediction Explanation: The student is given a table of winning heights for the men's Olympic pole vault competitions up to the year 2004. The student is asked to create a scatter plot and determine whether a linear model is appropriate. Then they need to find and graph the regression line, use the linear model to predict the height of the winning pole vault at the 2008 Olympics and compare with the actual winning height of 5.96 meters. Finally, they need to determine if it's reasonable to use the model to predict the winning height at the 2100 Olympics. Step by Step Solution: Step 1. Create a scatter plot of the winning heights and the corresponding years to determine the relationship between the two variables. Step 2. Based on the scatter plot, determine if the relationship appears to be linear or nonlinear. If the relationship is roughly linear, a linear model may be appropriate. Step 3. Compute the equation of the linear regression line that approximates the relationship between the winning heights and the corresponding years. The equation takes the form: y = a + bx, where y is the winning height, b is the slope of the line, x is the year, and a is the y-intercept. Step 4. Graph the regression line along with the scatter plot to visualize the fit of the line to the data. Step 5. Use the linear model to predict the height of the winning pole vault at the 2008 Olympics by substituting 2008 for x in the regression equation. Compare this prediction with the actual winning height of 5.96 meters. Step 6. Determine if the linear model is reasonable to use in predicting the winning height at the 2100 Olympics. Use the plot to assess whether the linear approximation will become unreasonable at some future year.