Question

(i) identify the variable(s) in the study, (ii) for each variable tell the type of variable (e.g., categorical and ordinal, discrete, etc.), (iii) identify the observational unit (the thing sampled), and (iv) determine the sample size. (a) A biologist measured the number of leaves on each of 25 plants. (b) A physician recorded the number of seizures that each of 20 patients with severe epilepsy had during an eight-week period.

Short answer

For the biologist's study, the variable is 'number of leaves', a discrete numerical variable, with the plant as the unit of observation and a sample size of 25. For the physician's study, the variable is 'number of seizures', also a discrete numerical variable, with the patient as the unit of observation and a sample size of 20.

Step-by-step solution

Identify Variables for Plant Study

In the biologist's study, the variable is the 'number of leaves on a plant'.

Determine Type of Variable for Plant Study

The variable 'number of leaves on a plant' is a discrete numerical variable as it is a count of leaves.

Identify Observational Unit for Plant Study

For the biologist's study, the observational unit is each individual plant.

Determine Sample Size for Plant Study

The sample size for the biologist's study is the number of plants observed, which is 25.

Identify Variables for Epilepsy Study

In the physician's study, the variable is the 'number of seizures' a patient had during an eight-week period.

Determine Type of Variable for Epilepsy Study

The variable 'number of seizures' is a discrete numerical variable because it is a count of occurrences.

Identify Observational Unit for Epilepsy Study

For the physician's study, the observational unit is each individual patient with severe epilepsy.

Determine Sample Size for Epilepsy Study

The sample size for the physician's study is the number of patients who were observed, which is 20.

Key concepts

Discrete Numerical Variable

In statistics, a discrete numerical variable is one that can take on a countable number of distinct values. This type of variable is best understood through examples, such as the one provided in our textbook exercise where a biologist measures the number of leaves on plants. The variable in question is the number of leaves, which is inherently quantitative and countable—a core characteristic of discrete variables.

The difference between a discrete variable and its counterpart, the continuous variable, is important to note. While a discrete variable like the number of leaves on a plant can be counted in whole numbers (1, 2, 3, and so forth), a continuous variable would be measured on an infinite scale, such as the length of a leaf, which can have an infinite number of potential values even between any two given points.

In the context of our exercise, knowing whether a variable is discrete or continuous is crucial for choosing the correct statistical methods for analysis. For instance, discrete variables often make use of different graphical representations, like bar graphs, and different summary statistics, such as mode or median, when it comes to investigating the data.

Observational Unit

The observational unit is essentially the entity that is being studied or observed. In a statistical study, each observational unit provides a data point, and the nature of this unit can vary widely depending on the study's focus. For example, in the biology study discussed in the textbook exercise, the individual plants acted as the observational units, because each plant represented a separate unit of observation.

It is critical to correctly identify the observational unit in any statistical study because it sets the stage for analyzing the data. If we confused our units of analysis, we would risk compromising the integrity and accuracy of our findings. To avoid such pitfalls, researchers must define what or whom they are studying—whether it be plants, as in the biologist's case, or patients with epilepsy in the physician's study—very clearly before they begin their work.

Understanding and being clear about the observational unit helps in ensuring that the data collected is relevant and that the conclusions drawn from the analysis reflect the actual relationships or patterns present in the data.

Sample Size

The sample size in a statistical study refers to the number of observational units included. It's a crucial component in research design, as the size of the sample can influence the reliability and accuracy of the study's results. In the biologist's study from the exercise, the sample size is 25 plants, while in the physician's study on seizures, it's 20 patients.

It's not just a matter of having a big enough sample size to achieve statistical significance, but it's also about balancing resources such as time and money. Researchers have to consider what is known as the power of the study, which describes the study's ability to detect an effect if there is one to be found. A sample too small may miss the effect entirely, while an unnecessarily large sample could waste resources.

Determining the correct sample size is often done through a power analysis, which estimates the minimum number of observations needed to detect a meaningful effect size with a desired level of confidence. In both example studies, the sample sizes chosen would have been based on considerations about the expected variability in measurements, the desired precision of the results, and the constraints of the study situation.