Name:
Name: ___________________________
AP Statistics - Residuals Notes & Worksheet
|x |y |[pic] |[pic] |
|2 |1 | | |
|3 |4 | | |
|5 |6 | | |
|6 |5 | | |
|7 |9 | | |
|9 |8 | | |
|10 |11 | | |
1. Plot the scatterplot for the points above.
2. Find the LSRL and correlation coefficient. (Round to 4 decimal places)
[pic] = ______________________________ r = _________________
3. Use the LSRL to calculate the predicted (fitted) value for each x-value. Fill in the chart above.
4. Calculate the residuals ([pic]) and fill in the chart above.
5. Create a residual plot by plotting a scatterplot of the 6. Create another residual plot by plotting
x-values on the horizontal axis and the residuals on the the [pic]-values on the horizontal axis and
vertical axis. the residuals on the vertical axis.
7. What do you notice about these two residual
plots?
8. Is the LSRL from question 2 an appropriate
model for this data? Explain.
One measure of the success of knee surgery is post-surgical range of motion for the knee joint following a knee dislocation.
Age 45 40 31 28 26 16 14 20 21
Range of Motion 205 137 133 122 135 135 108 120 127
9. Sketch a scatterplot of this data and calculate the LSRL.
Plot the LSRL on your scatterplot.
[pic] = ______________________________
10. Does there appear to be an influential point? IF so, which one?
11. Remove the point from your list. Sketch a scatterplot of this
data and calculate the LSRL. Plot the LSRL on your scatterplot.
[pic] = ______________________________ r = _____ r2 = ______
12. Would you consider this point influential? Explain.
13. Write interpretations, in context, for the slope, r & r2 for question 11.
14. Which of these measures would be considered resistant? (LSRL, correlation coefficient (r), or coefficient of
determination (r2))
-----------------------
_________________________ is the vertical distance that a point on a scatterplot is from the LSRL.
The sum of the residuals is ALWAYS equal to ______________________________.
Residuals = [pic]
x
Residuals
Residuals
[pic]
Residual Plots
• A scatterplot of the residuals plotted against the
x-values or the predicted or fitted values ([pic]).
• The purpose of a residual plot is to determine if the
model (equation) is an appropriate fit for the data.
o The residual plot should look like a random
scatter of points.
o If no pattern exists between the points in the
residual plot, then the model is appropriate.
o If a pattern does exist, then the model is not
appropriate for the data.
Coefficient of determination (r2)
• gives the proportion of variation in y that can be attributed to an approximate linear relationship between x & y
• remains the same no matter which variable is labeled x
In a regression setting, an outlier is a data point with a large residual.
An influential point is a point that influences where the LSRL is located. If removed, it will significantly change the slope of the LSRL.
Age
Range of Motion
Age
Range of Motion
Interpretations: (replace the underlined items with correct values or words in context)
Slope: For each unit increase in x, there is an approximate increase/decrease of b in y.
Correlation coefficient: There is a direction, strength, linear of association between x and y.
Coefficient of determination: Approximately r2% of the variation in y can be explained by the LSRL of x & y.
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