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Exercises 59 to 60 refer to the following setting. For their final project, a group of AP \(^{\otimes}\) Statistics students investigated the following question: "Will changing the rating scale on a survey affect how people answer the question?" To find out, the group took an SRS of 50 students from an alphabetical roster of the school's just over 1000 students. The first 22 students chosen were asked to rate the cafeteria food on a scale of 1 (terrible) to 5 (excellent). The remaining 28 students were asked to rate the cafeteria food on a scale of 0 (terrible) to 4 (excellent). Here are the data: $$ \begin{array}{lcccrc} &{1 \text { to 5 scale }} \\ \text { Rating } & 1 & 2 & 3 & 4 & 5 \\ \text { Frequency } & 2 & 3 & 1 & 13 & 3 \\ \hline & {0 \text { to 4 scale }} \\ \text { Rating } & 0 & 1 & 2 & 3 & 4 \\ \text { Frequency } & 0 & 0 & 2 & 18 & 8 \\ \hline \end{array} $$ $$ \text { Design and analysis }(4.2) $$ (a) Was this an observational study or an experiment? Justify your answer. (b) Explain why it would not be appropriate to perform a chi-square test in this setting.

Step by step solution

01

Identify Study Type

In part (a), we need to determine if the given scenario is an observational study or an experiment. In an observational study, researchers observe and measure outcomes without imposing any treatment or change. In an experiment, researchers deliberately impose treatments or changes to observe their effects. In this scenario, the students are imposing two different rating scales (1-5 and 0-4) on different groups of students to observe the impact on their responses. Thus, this qualifies as an experiment because the rating scale is being deliberately changed to investigate its effects.

02

Assess Appropriateness of Chi-Square Test

For part (b), we must evaluate why a chi-square test is not suitable here. A chi-square test is used to determine if there is a significant association between categorical variables or if observed frequencies in categories fit a specified distribution. In this context, the ratings from two different scales (1-5 and 0-4) are not truly comparable because each scale represents a different conceptual framework, even if numerically similar. Thus, the categorical nature and lack of commonality between these two scales prevent a meaningful application of the chi-square test, as they lead to non-comparable data distributions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Observation vs Experimentation

When we look at the study process, a key distinction is whether the approach taken is observational or experimental. In an observational study, the researcher simply observes the subjects without any interference. This means they document what naturally occurs without altering variables or imposing any new conditions. It's like being a silent audience member in a play.

In contrast, experimentation involves actively changing one variable to see if it causes an effect on another. Typically, the researcher imposes a treatment or condition on the subjects. For instance, our AP Statistics students decided to modify the rating scales from 1 to 5 and from 0 to 4 to assess their influence on the ratings given by students. Therefore, this study is an experiment. The students deliberately changed the scales, aiming to determine if it impacts the way subjects rate cafeteria food. This manipulation characterizes the study as an experiment.

Chi-Square Test

The chi-square test is a statistical tool used to examine if there is a significant association between categorical variables. It helps us understand if observed data deviate from the expected data under a given hypothesis. It's vital to ensure that the categories being compared are similar enough to yield meaningful results.

In our AP Statistics scenario, students were surveyed using two distinct rating scales. Specifically, one group rated using a 1 to 5 scale and the other using a 0 to 4 scale. Even though these seem similar, they represent different measurable frameworks. Each scale has additional nuances and impacts on data interpretation, which means they don't form a common basis for comparison. That makes the chi-square test inappropriate in this case. The scales create non-comparable categories, disrupting the assumptions needed for a chi-square analysis.

Survey Design

Executing a well-thought-out survey design is crucial to gathering reliable and meaningful data. In survey research, clarity, consistency, and unbiased questions are paramount. It starts with determining the aim of the survey, which in this context is to find out if the rating scale affects student responses about cafeteria food.

The students selected a simple random sample (SRS) from the school roster. This method is a robust way to ensure every participant has an equal chance of being chosen, which minimizes selection bias. The next step was to administer the survey questions with clear and straightforward scales. The survey design here cleverly implements two rating scales to explore their effect. But, it's essential to note that every element, from question wording to scale definition, should remain consistent within each subset to avoid confusing participants.

Rating Scales

Rating scales are commonly used tools in surveys that allow respondents to express their perception or evaluation neatly and quantitatively. Commonly, they offer a range, in this study from 1 to 5 or 0 to 4, that corresponds to a gradient of opinions from negative to positive.

Each rating scale presents unique calibration and interpretation, which affects the survey's outcomes. For example, a 1 to 5 scale may imply different strength or extremes of opinion than a 0 to 4 scale. While both aim to gauge satisfaction with the cafeteria food, their differing start and endpoints can shift perception slightly. Thus, the design using varied scales thoughtfully examines their impact on the data. However, for accurate comparisons across scales, adjustments or considerations regarding these inherent differences need to be made.