35. A student working on a physics project investigated the relationship between the speed and the height of roller coasters. The student collected data on the maximum speed, in miles per hour, and the maximum height, in feet, for a random sample of 21 roller coasters, with the intent of testing the slope of the linear relationship between maximum speed and maximum height. However, based on the residual plot shown, the conditions for such a test might not be met.
Based on the residual plot, which condition appears to have been violated?
(A) The errors are independent.
(B) The sum of the residuals is 0 .
(C) The expected value of the errors is 0 .
(D) There is a linear relationship between the response variable and the explatiable
(E) The variance of the response variable is constant for all values of the explanatory variable.

Question

35. A student working on a physics project investigated the relationship between the speed and the height of roller coasters. The student collected data on the maximum speed, in miles per hour, and the maximum height, in feet, for a random sample of 21 roller coasters, with the intent of testing the slope of the linear relationship between maximum speed and maximum height. However, based on the residual plot shown, the conditions for such a test might not be met.
Based on the residual plot, which condition appears to have been violated?
(A) The errors are independent.
(B) The sum of the residuals is 0 .
(C) The expected value of the errors is 0 .
(D) There is a linear relationship between the response variable and the explatiable
(E) The variance of the response variable is constant for all values of the explanatory variable.

Accepted Answer

∻Solution∻ ‖ a ⋮ The residual plot is a graphical representation that shows the differences between observed and predicted values of the dependent variable. ‖ ‖ b ⋮ If the variance of the residuals is not constant across all values of the explanatory variable, this suggests that the condition of homoscedasticity is violated. ‖ ‖ c ⋮ A residual plot that shows a pattern or systematic spread of residuals would indicate that the variance of the response variable is not constant. ‖ ***E*** ∻Key Concept∻ ⚹ Homoscedasticity: This is the condition that the variance of the residuals is constant across all levels of the explanatory variable. It is a key assumption in linear regression analysis. ⚹ ∻Explanation∻ ⚹ The residual plot can be used to check for homoscedasticity. If the residuals fan out or show a pattern, this indicates heteroscedasticity, meaning the variance of the residuals changes with the explanatory variable, violating the assumption of constant variance. ⚹

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