Find the residual values, and use the graphing calculator tool to make a residual plot.
| x | Given | Predicted | Residual |
|---|---|---|---|
| 1 | -3.5 | -1.1 |
| 2 | -2.9 | 2 |
| 3 | -1.1 | 5.1 |
| 4 | 2.2 | 8.2 |
| 5 | 3.4 | 11.3 |
Does the residual plot show that the line of best fit is appropriate for the data?
A. Yes, the points have no pattern.
B. No, the points are evenly distributed about the [tex]$x$[/tex] axis.
C. No, the points are in a linear pattern.
D. Yes, the points are in a curved pattern.
Answer (2)
The residuals were calculated using the given and predicted values to determine their differences, leading to a pattern indicative of a poor fit for the data. The calculated residuals suggest a trend, indicating that a linear model may not be appropriate. Therefore, the answer is: C. No, the points are in a linear pattern.
;
Calculate the residuals by subtracting the predicted values from the given values: Residual = Given − Predicted .
Observe the calculated residuals: -2.4, -4.9, -6.2, -6.0, -7.9.
Analyze the residual plot: The residuals show a decreasing trend as x increases, indicating a pattern.
Conclude that the line of best fit is not appropriate because the points in the residual plot are in a linear pattern: No, the points are in a linear pattern.
Explanation
Calculating 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. The formula for the residual is:
Residual = Given − Predicted
For the first data point, x = 1 :
Residual = − 3.5 − ( − 1.1 ) = − 3.5 + 1.1 = − 2.4
For the second data point, x = 2 :
Residual = − 2.9 − 2 = − 4.9
For the third data point, x = 3 :
Residual = − 1.1 − 5.1 = − 6.2
For the fourth data point, x = 4 :
Residual = 2.2 − 8.2 = − 6.0
For the fifth data point, x = 5 :
Residual = 3.4 − 11.3 = − 7.9
So, the residual values are -2.4, -4.9, -6.2, -6.0, and -7.9.
Understanding Residual Plots Now, let's analyze the residual plot. A residual plot is a graph where the x-values are plotted against the residual values. In this case, the x-values are 1, 2, 3, 4, and 5, and the corresponding residual values are -2.4, -4.9, -6.2, -6.0, and -7.9.
If the line of best fit is appropriate for the data, the residual plot should show no discernible pattern. The points should appear to be randomly scattered around the x-axis. If there is a pattern, such as a curved or linear pattern, it indicates that the line of best fit is not a good fit for the data.
Analyzing the Pattern Looking at the calculated residuals (-2.4, -4.9, -6.2, -6.0, -7.9), we can see that as the x-values increase, the residual values tend to become more negative. This suggests a pattern in the residual plot. The points are not randomly scattered; they appear to follow a decreasing trend. This indicates that the line of best fit is not appropriate for the data.
Final Answer and Conclusion Since the residual plot shows a pattern (the points are not randomly scattered), the line of best fit is not appropriate for the data. The points are not evenly distributed about the x-axis, and they do not have no pattern. The points are in a decreasing pattern, which is a type of linear pattern. Therefore, the correct answer is:
No, the points are in a linear pattern.
Examples
In weather forecasting, a line of best fit might be used to predict temperature changes over time. If a residual plot of the actual temperatures versus the predicted temperatures shows a pattern, it indicates that the model needs refinement to provide more accurate forecasts. By analyzing the residuals, meteorologists can improve their models and make better predictions.
