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Question:Charles uses a graphing calculator to find a quadratic regression model f for a given set of data. When he compares model f to an earlier regression model g for the same data, he determines that g more accurately models the data. Select all the true statements.
A. Function f likely had fewer residuals near the x-axis than function g.
B. Function f likely had more residuals equal to 0 than function g.
C. Function g had more residuals near the y-axis than function f.
D. Function g likely had a R<span id="MathJax-Element-1-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="2" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.8px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">22 value closer to 1 than function f.
E. Function g likely had all positive residual values.
Which answers are correct and why?
Residual analysis:Linear regression seeks to find a line of regression through one or more independent variables and a dependent variable by minimizing the squares of the residuals. The residual is a measure of the distance between a specific observation and the line of regression. The squares of the residuals are commonly evaluated post regression to determine the presence of heteroskedasticity or other issues with the regression model.
Answer and Explanation:Let's evaluate and discuss each statement to determine which are true:
A. Function f likely had fewer residuals near the x-axis than function g.
If a function includes more values near the "x"-axis this indicates heteroskedasticity. Picture a scatterplot for two variables, one dependent and the other independent, and ideally it is football shaped and shows either a positive or negative relationship between the two variables. In other words as one variable increases the other either increases or decreases and the observations are moving sway from or toward the "x"-axis. The distance between the line of regression and these observations is the residual so the function that has fewer residuals near the "x"-axis will be a better fit. Since function "g" was a better fit then it was the function with fewer residuals near the "x"-axis. This statement is false.
B. Function f likely had more residuals equal to 0 than function g.
The residual is the distance between the observation and the line of regression. If on a scatterplot the observations were in a straight line (i.e. a perfect fit) then there would be zero distance between the observations and line of regression so the function with the better fit will have more residuals equal to 0. In this case function "g" is the better fit so it had more residuals closer to 0. This statement is false.
C. Function g had more residuals near the y-axis than function f.
The reasoning for this statement is the same as given in statement A. The function with fewer residuals near the "y"-axis would be a better fit, or in this case function "g". This statement is false.
D. Function g likely had a R<span id="MathJax-Element-2-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="2" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.8px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">22 value closer to 1 than function f.
The value of R<span id="MathJax-Element-3-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="2" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.8px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">22 is an indicator of much of the variation in the independent variable is explained by the model. This value will be between 0 and 1 and is often represented as a percentage. For example a regression model with an R<span id="MathJax-Element-4-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="2" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.8px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">22 of 0.75 would be said to explain 75% of the variation in the independent variable. The function that fits better will have an R<span id="MathJax-Element-5-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="2" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.8px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">22 closer to 1 than a function with an R<span id="MathJax-Element-6-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="2" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.8px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">22 closer to 0. In this case function "g" would have an R<span id="MathJax-Element-7-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="2" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 16.8px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">22 closer to 1. This statement is true.
E. Function g likely had all positive residual values.
The linear regression model finds a line of regression that minimizes the squares of the residuals. This means there will be ideally as many observations above the line of regression as below the line of regression and residuals will be equally positive and negative when summed. Since function "g" was a better fit this means the sum of the residuals was closer to zero than function "f" so it could not have all positive residuals. This statement is false.
The only statement that is true is statement D.
