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Part (a): Recognize that the slope of the least-squares regression line represents the expected change in y associated with a 1 unit increase in x. 

Students generally did a reasonable job in responding to part (a) of this question, although not very many answered the question using the anticipated approach based on recognizing how the slope of the least-squares line could be used to answer this question. As a result, most students spent time calculating predicted values and finding the difference in order to answer this question.

The most common error in answering this question was made by students who realized that they needed to use the given equation of the least-squares line in some way, but did not use it appropriately. These students substituted 5 into the equation to obtain the predicted arm span for a person whose height was 5 cm! This is illustrated by the student responses below, which were scored as incorrect for part (a).

A few students did not know how to use the least-squares line to answer part (a). For example, consider the student response below. While this student has recognized that there is a positive relationship between height and arm span, the student was not able to provide a value for the expected difference, and this response was scored as incorrect for part (a).

Finally, even though the question specifically asks students to show their work, some responses did not show supporting work and were scored as only partially correct. The student response below would have been considered correct (assuming that the student rounded to the nearest cm) if supporting work had been included.

 Part (b): Calculate a predicted value and the value of a residual given the equation of the least-squares regression line. Interpret a residual as the difference between an observed y value and the corresponding predicted y value. 

Students had more difficulty responding to part (b) than the other parts of this problem and there were a number of relatively common student errors. For example, some students calculated predicted values, but did not calculate residuals or explicitly compare the predicted values with the given actual arm spans for Rhonda or Jane. This is illustrated in the following student response, which received a score of partially correct for part (b) because it is not clear what the student means by the statement that “her arm span length correlates more closely with the least-squares regression line.”

Many students correctly indicated that the prediction of Rhonda’s arm span was more accurate, but then provided an incorrect explanation. This error is illustrated in the following two student responses. In the first of these responses, the student calculates the residuals correctly but then provides an incorrect interpretation of those residuals. In the second response below, the provided justification does not include an explanation based on residuals or the distance of the points to the line. While it is true that the scatterplot and least-squares line indicate that arm span tends to be greater than height, this alone is not enough to guarantee that predictions would be more accurate for someone whose arm span is greater than his or her height. Each of these responses was scored as partially correct for part (b).

Some student responses indicated that they believed that it is not possible to tell which prediction would be more accurate. For example, consider the student response below. In this response, the student indicates that the line would not be more accurate for one or the other because there is a lot of scatter around the regression line. While that does imply that some prediction errors might be large, it is still possible to evaluate the arm span predictions for Rhonda and Jane and to determine which is closer to actual arm span.

It was also common for students to make errors in calculating the predicted values or the residuals. The following two student responses are typical of the responses of students making this error. Each of these responses were scored as partially correct for part (b).

Finally, some students provided an explanation that was found to be insufficient for a score of essentially correct for part (b). In the first of the two student responses below, the student explains how it would be possible to arrive at an answer, but gives no evidence that the steps described were actually carried out. This response was scored as partially correct for part (b). The second of the two responses below was scored as incorrect for part (b). Although it references the graph, there were no markings on the graph to indicate that the student has plotted the two relevant points and considered the distance of these points to the least-squares regression line.

Part (c): Recognize that the least-squares regression line should not be used to make predictions for x values that are far outside the range of the data used to determine the equation of the least-squares regression line.

Responses to part (c) that were not scored as essentially correct generally made one of two common errors. Some students correctly indicated that the least-squares regression line should not be used to make a prediction of Doug’s arm span, but did not provide an explanation that clearly indicated that the reason a prediction should not be made was that Doug’s height was outside the range of height in the data set used to determine the equation of the least-squares line. For example, in the student response below, the student notes that Doug is “abnormally tall” but does not compare Doug’s height to the heights in the data set. For this reason, this student response was scored as only partially correct for part (c).

The response below also indicated that the least-squares line should not be used to make a prediction, but justifies this by saying the equation is less precise for tall people. While this might be true, there is nothing in the given scatterplot to suggest that this is the case and it does not address the key issue of extrapolation beyond the range of the data. This response was also scored as only partially correct for part (c).

The following student responses were also scored as partially correct for part (c). Although they appear to be on the right track, it is not clear that they are looking at the actual range of values in the data set or just the scale on the axis. Because it is not clear that these students are addressing the issue of extrapolation beyond the range of the data, these responses were not scored as essentially correct.

The second common student error was made by students who did not recognize that the question was asking for a prediction for a height value that was far outside the range of the data set. Students making this error generally indicated that it was appropriate to use the least-squares line to make a prediction of Doug’s arm span. The four responses below all make this error. Some also include incorrect statements in the explanation. For example, the second response below states that the least-squares line “can be used for values outside of the chart.”  These four responses were all scored as incorrect for part (c).

Some students didn’t actually answer the question asked, and instead just used the equation of the least-squares line to predict Doug’s arm span. This error is illustrated by the following student response, which was also scored as incorrect for part (c).