Find the residual values and use the graphing calculator tool to make a residual plot.
| x | Given | Predicted | Residual |
|---|---|---|---|
| 1 | -2.7 | -2.84 |
| 2 | -0.9 | -0.81 |
| 3 | 1.1 | 1.22 |
| 4 | 3.2 | 3.25 |
| 5 | 5.4 | 5.28 |
Does the residual plot show that the line of best fit is appropriate for the data?
A. No, the points are in a curved pattern.
B. No, the points are evenly distributed about the $x$ axis.
C. Yes, the points are in a linear pattern.
D. Yes, the points have no pattern.
Answer (1)
Calculate the residuals by subtracting the predicted values from the given values.
The residuals are: 0.14, -0.09, -0.12, -0.05, 0.12.
Analyze the residual plot (or the residual values) to determine if there is a pattern.
Since the points are evenly distributed about the x axis, the line of best fit is appropriate. The answer is: No, the points are evenly distributed about the x axis.
Explanation
Calculate Residuals First, we need to calculate the residual values for each data point. The residual is the difference between the given value and the predicted value.
Residual for x = 1 For x = 1 , the residual is − 2.7 − ( − 2.84 ) = 0.14 .
Residual for x = 2 For x = 2 , the residual is − 0.9 − ( − 0.81 ) = − 0.09 .
Residual for x = 3 For x = 3 , the residual is 1.1 − 1.22 = − 0.12 .
Residual for x = 4 For x = 4 , the residual is 3.2 − 3.25 = − 0.05 .
Residual for x = 5 For x = 5 , the residual is 5.4 − 5.28 = 0.12 .
Analyze Residual Plot Now we have the following residual values: 0.14, -0.09, -0.12, -0.05, 0.12. A residual plot is a graph where the x-axis represents the x-values and the y-axis represents the residual values. If the points in the residual plot are randomly scattered around the x-axis, then the line of best fit is appropriate. If there is a pattern (e.g., a curved pattern), then the line of best fit is not appropriate.
Conclusion about the pattern In this case, without actually plotting the points, we can analyze the values. The residuals are: 0.14, -0.09, -0.12, -0.05, 0.12. These values do not seem to follow a specific pattern, and they are relatively small, scattered around zero. Therefore, it is reasonable to assume that the points are evenly distributed about the x-axis.
Examples
In data analysis, understanding residuals helps us assess the accuracy of our models. For example, if we're predicting house prices based on size, a residual plot can show if our model is systematically over or under-predicting for certain sizes. By analyzing the residuals, we can refine our model to make more accurate predictions, ensuring fair pricing and better investment strategies. This process is crucial in fields like real estate, finance, and economics, where accurate predictions are essential for making informed decisions.
