Question

A student study of the effects of caffeine asked volunteers to take a memory test 2 hours after drinking soda. Some drank caffeine-free cola, some drank regular cola (with caffeine), and others drank a mixture of the two (getting a half-dose of caffeine). Here are the 5-number summaries for each group's scores (number of items recalled correctly) on the memory test: $$\begin{array}{lcccccc}& n& \text{ Min }& \text{ Q1 }& \text{ Median }& \text{ Q3 }& \text{ Max }\\ \text{ No caffeine }& 15& 16& 20& 21& 24& 26\\ \text{ Low caffeine }& 15& 16& 18& 21& 24& 27\\ \text{ High caffeine }& 15& 12& 17& 19& 22& 24\end{array}$$ a) Describe the W's for these data. b) Name the variables and classify each as categorical or quantitative. c) Create parallel boxplots to display these results as best you can with this information. d) Write a few sentences comparing the performances of the three groups.

Short answer

Data compares caffeine effect on memory: no/low caffeine recall higher than high caffeine.

Step-by-step solution

Identifying the W's

The "W's" are who, what, when, where, why, and how, which describe the context of the dataset. - **Who**: The participants in the study, which are the volunteers who took the memory test. - **What**: The test scores on the memory test, which are quantitative data showing the number of items recalled correctly. - **When**: The timeframe is not specified, but the test was conducted 2 hours after soda intake. - **Where**: Not specified in the problem. - **Why**: To study the effect of caffeine on memory recall. - **How**: By administering a memory test after consumption of soda with varying caffeine levels.

Classifying Variables

- The variable related to participants' test scores is **quantitative**, as it represents numerical data indicating the number of items recalled correctly. - The caffeine level (no caffeine, low caffeine, high caffeine) is a **categorical** variable, as it is a type of classification into distinct groups based on caffeine intake level.

Creating Parallel Boxplots

With the given 5-number summaries, we can construct boxplots. These summarize distributions by the five key data points: minimum, first quartile $Q1$, median, third quartile $Q3$, and maximum. For each group:- **No caffeine**: Min = 16, Q1 = 20, Median = 21, Q3 = 24, Max = 26.- **Low caffeine**: Min = 16, Q1 = 18, Median = 21, Q3 = 24, Max = 27.- **High caffeine**: Min = 12, Q1 = 17, Median = 19, Q3 = 22, Max = 24.In a parallel boxplot, these values would define the range, center, and spread of the boxes and whiskers for each group's data.

Comparing Group Performances

The boxplots show that the median scores are highest for the "No caffeine" and "Low caffeine" groups (both medians at 21) compared to "High caffeine" (median at 19). The "No caffeine" group has less spread and the highest minimum score (16), indicating consistently higher performance relative to other groups. "High caffeine" scores have the lowest median and minimum, suggests that high caffeine might negatively impact memory recall performance.

Key concepts

Boxplot

A boxplot, also known as a whisker plot, is a graphical representation used in descriptive statistics to display the distribution of data based on a five-number summary. These numbers include the minimum, the first quartile (Q1), the median, the third quartile (Q3), and the maximum. Boxplots are incredibly helpful for visualizing the spread and skewness of the data.
They can also highlight outliers that might exist in the dataset. Each part of the boxplot tells a story:

• The **box** itself represents the interquartile range (IQR), the central 50% of the data.
• The **line inside the box** marks the median, giving a sense of the central tendency of the data.
• The **"whiskers"** extend from the quartiles to the minimum and maximum values, showing the overall spread of the data.

Using boxplots for the caffeine study provides a quick visual comparison of memory recall performance among the three different groups with varying caffeine intake levels.

Quantitative Variable

A quantitative variable represents data that can be measured and expressed numerically. In contrast to categorical variables, which simply classify information, quantitative variables provide insights into the magnitude of differences and allow for complex mathematical comparisons.
In the caffeine study, the number of items recalled correctly by participants acts as a quantitative variable.
A few characteristics of quantitative variables include:

• They can be subjected to mathematical operations like addition and multiplication.
• They allow for a wide range of statistical analyses, such as calculating means, medians, and standard deviations.
• They provide a deeper understanding into variations in the dataset.

Quantitative variables are crucial in the caffeine study because they enable the comparison of test scores across the three groups, thereby assessing the potential impact of caffeine intake on memory.

Categorical Variable

Categorical variables categorize data into distinct, separate groups without a logical order. They are not numerical and do not engage in arithmetic operations, but they are essential for differentiating the data based on certain characteristics.
In the context of the caffeine study, the caffeine levels—no caffeine, low caffeine, and high caffeine—are considered categorical variables. These labels classify the participants into distinct groups depending on their intake of caffeine.
Using categorical variables in descriptive statistics often involves counting or grouping individuals into categories, simplifying complex data into more understandable partitions.
Understanding the impact of categorical variables, like those in the caffeine study, helps in identifying patterns or trends associated with classifications like caffeine intake levels.

Caffeine Study

The caffeine study investigates how different levels of caffeine intake affect memory recall in individuals. This research entails assigning participants to various groups based on their caffeine intake and evaluating their memory performance through a standardized test.
This study is crucial, as caffeine is one of the most widely consumed stimulants worldwide, with potential impacts on cognitive processes like memory, focus, and alertness. Key elements of the caffeine study include:

• Participants: Volunteers subjected to memory tests after consuming varying levels of caffeine.
• Method: A memory test conducted two hours post-intake of soda (caffeine-free, low caffeine, or high caffeine).
• Outcome: The number of items each participant recalls correctly serves as the primary measure of memory performance.

This simple design allows researchers to analyze whether caffeine enhances or detracts from cognitive capabilities, thereby influencing outcomes, such as memory recall.