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Part (a) Use categorical data that have been summarized in a frequency table and a bar graph to answer a statistical question

Most students provided correct responses to part (a) of this question. Responses that were not scored as essentially correct for part (a) generally made one of three common mistakes. The most concerning of these mistakes is illustrated by the following two student responses. In these responses, the student interprets the frequencies as if they were numerical data values and compute summary statistics only appropriate for numerical data (such as the mean or median). This represents a conceptual error in thinking about the underlying data, which is categorical. In the first response below, the range, median and mean of the 7 given frequencies have been computed, and these measures have no meaning in the context of categorical data. In the second response below, the student says “the average is online videos,” probably basing this statement on the fact that the frequency of 15 is closest to the average of 14.3 obtained when the frequencies are averaged. These two responses were both considered incorrect for part (a).

A second error that occurred in a number of papers was most likely the result of not reading the question carefully. The question asks for two statements that address the question “What do middle school students like to talk about with their friends?” Some students did not provide statements and just gave a poorly communicated answer to the question itself. This is illustrated by the following student response, which was scored as incorrect for part (a).

The third common error in part (a) occurred in responses that actually did provide two correct statements, but then went on to provide one or more additional statements that were not correct or that could not be inferred from the given table or bar graph. Depending on the error in the incorrect statement, such responses are considered to be either partially correct or incorrect for part (a). For example, the following response was scored as only partially correct for part (a) because the statements about “mostly girls do this” and “most boys” do not follow from either the table or graph and are based on personal opinion.

The following response includes two correct statements, but then follows them with an “extra” statement that is incorrect and is an example of the conceptual error previously described where frequencies are treated as the data. For this reason, this response was scored as incorrect for part (a).

Parts (b) and (c)  Understand that generalizing from a sample to a larger population is reasonable only if the sample is representative of the population.

Parts (b) and (c) both address the concept of sampling variability and when it is reasonable to generalize from a sample to a larger population.

For a correct response in part (b), students could either indicate that they would expect the results to be similar in a different sample as long as the samples were random samples or that they would not expect the results to be similar because it was not stated that the samples were randomly selected. In either case, the explanation includes a reference to the way the samples were selected. One common error in responding to part (b) was to indicate that the results for the two samples would be similar, but not justifying this by addressing the role that the sampling method plays. The following four responses illustrate this error by their failure to acknowledge the role of sampling method.

A different, but also concerning, common conceptual error is illustrated in the following student response to part (b). This response illustrates the error of believing that any sample is representative as long as the sample size is large. This response was scored as incorrect for part (b) and is typical of student responses making this error.

In part (c), the most common conceptual error was failure to recognize that even with a large random sample, it is only appropriate to generalize from a sample to the population from which the sample was selected. This is illustrated by the following three responses that all make indicate that results would be similar in a different city and justify this response by arguing that middle school students in other cities would talk about the same things. While the first response points out a reason why the result at a different school might be different (“If it is a high excelling school they might talk more about books than the opposite gender”), this statement is not linked to the sample being selected in a different city.