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Part (a): Identify the population to which it is reasonable to generalize conclusions based on sample data.

While most students were able to identify the appropriate population for the sampling method described in part (a), there were a number of relatively common student errors. The most common student error was failure to distinguish between the population of library card holders and those that are frequent users of the library. Responses making this error were scored as partially correct. This error is illustrated by the two student responses below.

Other students indicated that the results could not be generalized and only applied to the 100 people surveyed, even though the 100 people were randomly selected from a list of library card holders. The student response below was scored as incorrect for part (a), and illustrates this conceptual error.

Other students specified a population that was not related to library card holders. Each of the four student responses that follow make this error and were scored as incorrect for part (a). Based on the way in which the sample was selected (method 1), it would not be reasonable to generalize to any of the groups described in these responses.

There were also a surprising number of students who did not understand the question and so chose to answer a different question! For example, see the following two student responses to part (a).

Part (b): Create an appropriate graphical display.

By far the most common error in part (b) was the failure to take the different sample sizes into account when constructing the graphical display. Because students were asked to create a graph that could be used to compare the response distributions for the two surveys, it is necessary to use either relative frequencies or percentages for the two possible responses when constructing the graphical display. Many students did not do this and constructed graphs using the actual frequencies. The following two student responses illustrate this error and were each scored as only partially correct for part (b). Notice also that the first response does not include an appropriate label for the vertical axis and that the second response is also missing this label and does not use the same scale for both graphs.

A less frequent error on part (b) occurred in responses which did not recognize the categorical nature of the survey data and so attempted to create a graphical display more suited for numerical data (such as a dotplot or a scatterplot). The two student responses that follow are typical of the kinds of graphs produced by students making this error.

Part (c): Identify bias introduced by a sampling method that does not use random selection.

There were several common student errors in answering part (c). The most common of these errors involved failure to recognize that method 2 was unlikely to produce a sample that was representative of library card holders, and that this would result in different response distributions for the two surveys. Many students instead focused on the fact that the sample sizes were different. Although the sample sizes were different, it is unlikely that this alone would result in response distributions in which the proportions of yes responses were as different as what was observed. The following three responses are typical of those that made this error.

A similar error was made by students who believed that the observed difference was probably just the result of sample to sample variability and did not take the different sampling methods into account. For example, see the student response below which was scored as incorrect for part (c).

There were two other common errors that resulted in scores of partially correct for part (c). These errors were either failing to make a comparison of the two methods or recognizing the difference in the two sampling methods, and failing to explicitly indicate how this difference would impact the response distributions for the two surveys. A response that was scored partially correct because it did not compare the two sampling methods is shown below. It mentions something about method 1 that distinguishes it from method 2, but does not actually make the comparison or address how this would affect the response distributions.

 The following two student responses are typical of those that were scored as only partially correct for part (c) because they do not specify how the difference in the methods is expected to affect the response distributions.