Chapter 11: The R.M.S. Error for Regression Errors: A has a large positive error B has a large positive error C has a negative error D has a negative error E has a positive error
The r.m.s. error is the r.m.s. size of the errors. The r.m.s. error measures how good a prediction is. It says how large the errors are likely to be. To calculate the r.m.s. error use the following shortcut: r.m.s. error = √ (1 – r 2 ) (SD Y )
Example 1: For the men aged 18-24 in the HANES sample, the relationship between height and systolic blood pressure can be summarized as follows: Average height ≈ 70”, SD ≈ 3” Average b.p. ≈ 124mm, SD ≈ 14mm r = -0.2 Estimate the average blood pressure of men who were 6 feet tall. a) Find the r.m.s. error of the prediction b)
If the scatter diagram is football-shaped, the r.m.s. error is like an SD for the regression line. 68% of the dots fall between the line ± 1 r.m.s. error 95% of the dots fall between the line ± 2 r.m.s. errors
One r.m.s. error up and down Two r.m.s. errors up and down 68% 95%
If the scatter diagram is football-shaped, the r.m.s. error says how far a typical point is above or below the regression line. It gives us a give-or-take number for our estimates. Example 2: Midterm: ave = 65 SD = 16 r = 0.7 Final: ave = 60 SD = 10 Estimate the final exam score for someone who got 81 on the midterm and put a give-or-take number on your estimate.
If the scatter diagram is football-shaped, the r.m.s. error can be used like an SD for the regression line. Approximately 95% of the points will be between regression estimate – 2(r.m.s. error) and regression estimate + 2(r.m.s. error) Example 3: Midterm: ave = 65 SD = 16 r = 0.7 Final: ave = 60 SD = 10 Estimate the final exam score for someone who got 81 on the midterm. Would you be surprised to hear that the student scored 70? How about 77? 60?
Residuals The residual says how far the point is above or below the line. To see if the scatter diagram is football-shaped, we plot the residuals
Residual plots make it easier to see if the scatter diagram is football-shaped. Is this one football- shaped?
If the residual plot has a pattern, the regression might not be appropriate.
A football-shaped scatter diagram is said to be “________________”. A scatter diagram that has more variability on one side is said to be “_______________”. Which is this?
The r.m.s. error is only appropriate for ____________ scatter diagrams. If you don’t know what the scatter diagram looks like, it is dangerous to do the regression. In this case, you have to assume that it is football-shaped and if this assumption is incorrect, your answers may not be accurate.
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