Question

Athletes at the 2018 World Figure Skating Championships are evaluated in two rounds of competition: the Short Program, and the Free Skate. The sum of the scores for each skater are then used to determine the medalists. For each of the 24 male skaters at the 2018 World Championship, the scores for each round are displayed in a scatterplot. A graph titled Short Program versus Free Skate has free skate on the x-axis, and short program on the y-axis. A line of best fit goes through (160, 85) and (200, 100). A graph titled Residuals versus Free Skate has free skate on the x-axis, and residual on the y-axis. Points are scattered throughout the graph. Based on the scatterplot and the residual plot, is a linear model appropriate for predicting the Short Program score from the Free Skate score? A. A linear model is appropriate because the residuals are centered about zero. B. A linear model is not appropriate because there are outliers in the scatterplot. C. A linear model is not appropriate because the residual plot does not show a linear pattern. D. A linear model is appropriate because the scatterplot is roughly linear, and the residual plot does not show a clear pattern.

Answer 1

The residual plot reveals a non-random pattern, indicating a lack of homoscedasticity. While the scatterplot suggests linearity, the systematic pattern in the residuals suggests that a linear model may not be the best fit for predicting Short Program scores from Free Skate scores. Option C accurately captures this deviation from the assumptions of linear regression.Thus option c is the correct option.

Step-by-step explanation:

In the scatterplot, a line of best fit is drawn through (160, 85) and (200, 100), suggesting a linear relationship between the Short Program and Free Skate scores. However, the appropriateness of a linear model also depends on the residual plot. The residual plot is a crucial diagnostic tool for assessing whether the residuals (the differences between the observed and predicted values) exhibit a systematic pattern.

In this case, the residual plot does not show a random scatter of points around the horizontal axis, indicating a lack of homoscedasticity. Instead, there may be a discernible pattern in the residuals as we move along the x-axis, suggesting that the linear model might not be the best fit for the data. This violates one of the assumptions of linear regression, which assumes constant variance in the residuals.

While option A suggests that a linear model is appropriate if the residuals are centered about zero, this is not a sufficient condition. The key consideration is whether the residuals exhibit a random and unstructured pattern in the residual plot. In this scenario, the absence of a linear pattern in the residual plot (option C) is a more accurate assessment, indicating that a linear model is not the most suitable for predicting the Short Program score from the Free Skate score.

Therefore option c is the correct option.

Answer 2

A linear model is deemed appropriate when the scatterplot shows a linear relationship and the residuals display a random scatter centered around zero. Based on the patterns indicated, a linear model appears suitable for predicting scores in this case.

Step-by-step explanation:

The question asks whether a linear model is appropriate for predicting the Short Program score from the Free Skate score based on a scatterplot and a residual plot from the 2018 World Figure Skating Championships. In the scatterplot showing Short Program scores versus Free Skate scores, there is a line of best fit that goes through the points (160, 85) and (200, 100). The residual plot compares the residuals or errors on the y-axis to the Free Skate scores on the x-axis. If the residual plot displays a random scatter of points centered around zero without any discernible pattern, this indicates that the errors do not have a systematic trend and a linear model may be appropriate.

Answer choice D is correct, saying that a linear model is appropriate because the scatterplot is roughly linear, and the residual plot does not show a clear pattern. This implies that the line of best fit does capture the trend in the data well and that the residuals are randomly distributed, which are two assumptions for linear regression to be suitable. Therefore, the scatterplot and the absence of clear patterns or trends in the residual plot suggest that a linear model could accurately predict scores.