Question Browser

Part (a)
Identify what type of data (categorical or numerical) will result from a survey question.

Some students were not able to distinguish between categorical data and numerical data and indicated that the responses to the fast food question would be numerical. This was the most common error in answering part (a). The following student response is typical of those given by students making this error.

Other students did not understand the question or did read the question clearly and responded by indicating the two possible response categories (yes, no) rather than the data type of categorical or numerical. This error is illustrated in the following student response.

Part (b)
Identify the value of a row relative frequency that would be consistent with no association.

In part (b), a common misconception was that no association exists between gender and eating fast-food at least once per week only when the response percentages are 50% for each possible response category. Students making this error focused on dividing the responses up evenly among the possible response categories. The following two student responses are typical of those that made this error. Both of these responses were scored as incorrect for part (b).

Because the context for this problem results in two possible categories for each variable (Male/Female and Yes/No), the specific source of students’ misconception about 50% is unclear: is the 50% because of the two gender categories or the two fast-food categories? Had the context allowed for more than two response categories for the fast food question (such as “Always,” “Sometimes,” “Seldom,” or “Never”) then it would be easier to identify the cause of confusion based on whether the student responded with 25% or 50%.  

Some students believed that the 54% who reported eating fast food at least once per week would need to be divided evenly between males and females if there was no association, overlooking the important fact that there were more males than females in the group surveyed. This incorrect reasoning is illustrated by the following student response which was scored as incorrect for part (b).

Some student responses correctly indicated that the percentage would be 54%, but provided an explanation that was judged to be weak or incomplete. The two student responses that follow were both scored as only partially correct for this reason.

Many responses to part (b), such as the two below, focused on the expected count for males rather than the percentage. While responses like these show some understanding of the concept of independence between two categorical variables, they do not directly answer the question asked and so were scored as only partially correct for part (b).

Another error made by students in answering part (b) is illustrated by the following response. This student also indicates some understanding of the concept of independence, but does not make use of the information provided in the stem of the question and makes no reference to 54%. This response was considered partially correct for part (b).

Finally, some students did not base their answers to part (b) on the given information and responded based on personal opinion. This is illustrated in the following two student responses, each of which was scored as incorrect for part (b).

Part (c)
Calculate row relative frequencies for a given two-way table.

In part (c), many students gave the (correct) joint percentages rather than the row relative percentages needed for assessing association. The student response below is typical of those making this error. 

Another common error made by students was reporting the marginal percentages of males and females surveyed rather than the row relative percentages, as illustrated by the response below.

Part (d)
Determine if there is evidence of association between two categorical variables.

In part (d), many responses indicated that there is an association between gender and eating at a fast-food restaurant at least once a week but had justifications that were considered too weak to be marked as essentially correct. For example, in the response below, the justification omits the correctly calculated percentages and makes an implicit argument. Because there was no explicit comparison of the percentages, this response was scored as partially correct for the combined parts (c) and (d). 

In the response below, the justification is that “males eat more fast food than females” which is not exactly what is being measured, i.e. this response focuses on the amount of fast food eaten rather than the categorical response to the question about whether fast food is eaten at least once per week. While this or a similar answer would be scored as essentially correct if an explicit, appropriate comparison of percentages was made, simply listing the percentages together is not sufficient for a comparison.

In part (d), instead of justifying their answer using parts (b) and (c), many students answered based on their personal experiences without appealing to any of the given or calculated information. While the phrasing in the student response below may be ambiguous enough to allow the reader to infer the correct meaning, without further clarification this response is not correct. There is a tendency for students inexperienced with statistics to reason based on their personal beliefs and experiences without regard to data; the response below is consistent with this reliance on personal beliefs.

Other students focused on the different number of males and females surveyed and erroneously indicated that conclusions made when the sample sizes are different might be wrong (because of the sample size discrepancy). Two examples of this are given below.

Similar to a misconception that also surfaced in answers to part (b), some students compared the percentages to 50% rather than 54%. This illustrates the misconception that the responses must be split 50/50 if there is no association between gender and eating fast-food at least once per week. In fact, no association exists when the percentage of males eating fast food at least once per week and the percentage of females eating fast food at least once per week is the same as the percentage for the overall group, which in this case happens to be 54%. The response below indicates this misconception.