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Exercise

Estimating the best fit

Just as we used the mean and standard deviation to summarize a single variable, we can summarize the relationship between these two variables by finding the line that best follows their association.

Use the following function to select the line that you think does the best job of going through the cloud of points.

plot_ss(x = mlb11$at_bats, y = mlb11$runs,
        x1, y1, x2, y2)

This function will first draw a scatterplot of the first two arguments x and y. Then it draws two points (x1, y1) and (x2, y2) that are shown as red circles. These points are used to draw the line that represents the regression estimate. The line you specified is shown in black and the residuals in blue. Note that there are 30 residuals, one for each of the 30 observations. Recall that the residuals are the difference between the observed values and the values predicted by the line:

$$e_i = y_i - \hat{y_i}$$

The most common way to do linear regression is to select the line that minimizes the sum of squared residuals. To visualize the squared residuals, you can rerun the plot_ss() command and add the argument showSquares = TRUE.

plot_ss(x = mlb11$at_bats, y = mlb11$runs,
        x1, y1, x2, y2, showSquares = TRUE)

Note that the output from the plot_ss() function provides you with the slope and intercept of your line as well as the sum of squares.

Instructions

100 XP
  • Using plot_ss for at_bats and runs, choose a line that does a good job of minimizing the sum of squares. To help you get started, two (poorly) estimated points are already set.
  • Run the function several times, playing around with the coordinates of the red points. See if you can minimize your sum of squares.