Getting Started 
Misconception/Error The student does not appear to understand the distinction between correlation and causation. 
Examples of Student Work at this Level The student concludes that studying more causes examination scores to rise because as the amount of time spent studying increases, scores on the test also increase. The student does not appear to understand that causation cannot be inferred from correlational data.
Alternatively, the student concludes that studying more will not cause examination scores to increase for reasons unrelated to the distinction between correlation and causation. For example, the student says:
 The scores do not consistently rise.
 The study is flawed.
 The scores are not that different from each other.Â

Questions Eliciting Thinking On what basis did you make your decision?
Is it possible that there are other factors besides the time spent studying during the final five days that influenced exam scores?
Is it possible that the kinds of scores students received on their math tests in the past influence how long they study for the next math test? 
Instructional Implications Provide instruction on how causation can be established (e.g., by conducting a carefully designed experiment in which variables are controlled). Distinguish between an observational study and an experiment. Clarify that observational studies can only provide evidence of a correlation between two variables while welldesigned experiments test for a causal link between variables. Ask the student to look for examples in the real world (e.g., in news reports, newspapers, and magazines) for assertions of causality and to assess whether a causal conclusion appears warranted given the nature of the data collection.
Explain that the set of data provided is an example of correlated data since it comes from an observational study. Emphasize that correlation does not imply causation. Explain that when two variables, A and B, are correlated, it might be the case that A causes changes in B, B causes changes in A, or some third variable is influencing both A and B together. Provide the student with simple examples of correlated variables and explain why causation cannot be inferred (e.g., height and reading scores of students of a certain age are correlated since students who are five feet tend to read better than students who are three feet tall). Ask the student if this means that height is the cause of reading ability and to consider alternative explanations. Ask the student whether there could be a lurking variable that could affect both the number of hours studied during the final five days and the examination scores.
Consider implementing other MFAS tasks for SID.3.9. 
Making Progress 
Misconception/Error The student does not appeal to the correlational nature of the association in explaining why the inference is not supported. 
Examples of Student Work at this Level The student appears to understand that the association between studying duration and examination scores is not causal. However, the student does not clearly explain in terms of the distinction between correlational associations and causality. For example, the student says that one cannot infer that studying longer causes an increase in exam scores because:
 There does not seem to be enough evidence to support the conclusion.
 Studying might be a cause but there might be another cause.

Questions Eliciting Thinking What else is needed to support the conclusion?
Is there a correlation between studying duration and examination scores? What could be a reason for the correlation?
Does correlation imply causation? 
Instructional Implications Review the distinction between correlation and causality. Model using statistical terminology, such as correlation and causation, when explaining why one variable might not be a cause of another. Ask the student to reframe his or her explanation using statistical terminology. Introduce the concept of a lurking variable and ask the student whether there could be a lurking variable in the situation provided.
Consider implementing other MFAS tasks for SID.3.9. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student says that one cannot infer that studying longer causes examination scores to rise because:
 The data are from an observational study, so a causal conclusion is not justified although there is a trend that suggests a strong positive relationship between the number of hours studied and the final examination scores.
 The variables appear to be correlated, but the existence of even a strong correlation does not, by itself, prove causation.

Questions Eliciting Thinking Is it possible that studying longer causes examination scores to rise?
Why might those who studied less than two hours do better as a group than those who studied three hours? 
Instructional Implications Ask the student to use technology to find the maximum possible correlation coefficient that is consistent with the data. 