Question

For each of the following settings in Exercises \(2.1 .1-2.1 .5\) (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 paleontologist measured the width (in \(\mathrm{mm}\) ) of the last upper molar in 36 specimens of the extinct mammal Acropithecus rigidus. (b) The birthweight, date of birth, and the mother's race were recorded for each of 65 babies.

Short answer

For exercise (a), the variable is the molar width, which is continuous and quantitative, the observational unit is the specimen of Acropithecus rigidus, and the sample size is 36. For exercise (b), the variables are birthweight (continuous, quantitative), date of birth (interval), and mother's race (categorical, nominal), the observational unit is the baby, and the sample size is 65.

Step-by-step solution

Identify Variables for (a)

The variable for exercise (a) is the width of the last upper molar (in mm).

Determine Variable Type for (a)

The type of variable measured is continuous and quantitative, because it represents a measurement that can take on any value within an interval and is expressed in numbers.

Identify Observational Unit for (a)

The observational unit in exercise (a) is each specimen of the extinct mammal Acropithecus rigidus.

Determine Sample Size for (a)

The sample size in exercise (a) is 36 specimens.

Identify Variables for (b)

The variables for exercise (b) are birthweight (in grams), date of birth, and the mother's race.

Determine Variable Types for (b)

The type of the birthweight variable is continuous and quantitative. The date of birth is an interval variable because it represents points in time. The mother's race is a categorical and nominal variable, because it places each individual into a category without a natural order.

Identify Observational Unit for (b)

The observational unit in exercise (b) is each baby for whom the data were recorded.

Determine Sample Size for (b)

The sample size in exercise (b) is 65 babies.

Key concepts

Variable Identification

Understanding the variables in a study is a pivotal first step when interpreting or conducting research. A variable is anything that can take on different values across individuals or objects being studied. For example, in a study where a paleontologist measures the molars of Acropithecus rigidus, the width of the molar is the variable. Similarly, for an investigation on newborn babies, variables include birthweight, date of birth, and the mother’s race. Each of these can vary from one baby to another. Identifying variables correctly allows for proper analysis later in the study.

Variable Types

Variables come in various types, and classifying them accurately is crucial. Continuous variables, such as molar width or birthweight, can take on an infinite number of values within a range. Quantitative variables are measured numerically, like the width in millimeters or weight in grams. Interval variables, such as the date of birth, express data along a scale with equal intervals without a true zero point. On the other hand, categorical variables like the mother's race fall into distinct groups without a numerical value or order. Understanding these distinctions helps researchers choose the right statistical tools for analysis.

Observational Unit Determination

An observational unit is the entity from which data are collected. In the case of the paleontologist study, each specimen of Acropithecus rigidus represents an observational unit. For the study on babies, each baby is an observational unit. Determining the observational unit is essential for clarity on what exactly is being measured or observed, helping prevent data confusion and ensuring that the correct analysis methods are applied.

Sample Size Calculation

The sample size—the number of observations in a study—significantly impacts the confidence one can have in the results of that study. In practice, larger sample sizes tend to provide more reliable data, albeit with higher data collection costs. For statistical inference purposes, it's vital to know the sample size to assess the study's power and accuracy. In the examples given, the paleontologist's study had a sample size of 36 specimens, which was critical for ensuring sufficient statistical power and the ability to detect a true effect if one exists. The study on newborns had a sample size of 65, allowing for robust analyses of patterns or relationships between the various variables collected.