Consider the table showing the given, predicted, and residual values for a data set.
\begin{tabular}{|c|c|c|c|}
\hline$x$ & Given & Predicted & Residual \\
\hline 1 & -1.6 & -1.2 & -0.4 \\
\hline 2 & 2.2 & 1.5 & 0.7 \\
\hline 3 & 4.5 & 4.7 & -0.2 \\
\hline 4 & 6.1 & 6.7 & -0.6 \\
\hline
\end{tabular}
Which point would be on the residual plot of the data?
A. $(1,-1.6)$
B. $(2,1.5)$
C. $(3,4.5)$
D. $(4,-0.6)
Answer (2)
The residual plot consists of points (x, residual).
Identify the residual values for each x value from the table.
The points on the residual plot are (1, -0.4), (2, 0.7), (3, -0.2), and (4, -0.6).
The point that matches one of the given options is ( 4 , − 0.6 ) .
Explanation
Understanding Residual Plots We are given a table with x values, given values, predicted values, and residual values. The goal is to identify which point would be on the residual plot of the data. A residual plot consists of points where the x-coordinate is the original x value and the y-coordinate is the residual value.
Identifying Points on the Residual Plot From the table, we can extract the following points for the residual plot:
For x = 1 , the residual is − 0.4 , so the point is ( 1 , − 0.4 ) .
For x = 2 , the residual is 0.7 , so the point is ( 2 , 0.7 ) .
For x = 3 , the residual is − 0.2 , so the point is ( 3 , − 0.2 ) .
For x = 4 , the residual is − 0.6 , so the point is ( 4 , − 0.6 ) .
Comparing with Given Options Now we compare these points with the given options:
( 1 , − 1.6 ) - This is not on the residual plot. ( 2 , 1.5 ) - This is not on the residual plot. ( 3 , 4.5 ) - This is not on the residual plot. ( 4 , − 0.6 ) - This is on the residual plot.
Final Answer Therefore, the point ( 4 , − 0.6 ) would be on the residual plot of the data.
Examples
Residual plots are used in regression analysis to assess the appropriateness of a linear model. For example, if you are trying to predict house prices based on square footage, a residual plot can help you determine if a linear model is a good fit. If the residuals are randomly scattered around zero, a linear model is likely appropriate. If there is a pattern in the residuals, a non-linear model might be more suitable. This helps in making more accurate predictions and understanding the relationship between variables.
The point that would be on the residual plot is (4, -0.6), as it corresponds to one of the calculated residuals from the data table. Other options do not represent residuals. Therefore, the answer is option D: (4, -0.6).
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