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Chapter 2: Problem 184

A researcher claims to have evidence of a strong positive correlation \((r=0.88)\) between a person's blood alcohol content \((\mathrm{BAC})\) and the type of \(\mathrm{alco}-\) holic drink consumed (beer, wine, or hard liquor). Explain, statistically, why this claim makes no sense.

Short Answer

Expert verified
The researcher's claim of having evidence of a strong positive correlation between a person's BAC and the type of alcoholic drink consumed doesn't make sense, because correlation necessitates that both variables are continuous and from either interval or ratio scale. There's no ordering to the categories (beer, wine, hard liquor) therefore, does not make sense to compute a correlation between a nominal and a ratio variable.

Step by step solution

01

Understand the Variable Types

The type of alcoholic drink (beer, wine, or hard liquor) is a nominal variable, because there's no intrinsic ordering to the categories. The BAC is ratio level data since it has a clear definition of zero.
02

Understand the Correlation Coefficient

Correlation is a measure of linear association between two continuous variables, and makes no sense when applied to nominal variables, because it requires the variability in the data to be from an interval or ratio scale. The correlation coefficient (\(r\)) quantifies the strength and direction of the linear relationship between the two variables.
03

Explain Why the Claim Makes No Sense

The claim that there's a strong correlation between a nominal variable and a ratio variable doesn't make sense, because the correlation coefficient requires both variables to be continuous and from either interval or ratio scale. In other words, one cannot sensibly compute a correlation coefficient between BAC (continuous ratio variable) and type of drink (nominal variable).

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Key Concepts

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

Nominal Variables
Nominal variables represent categories that do not have a specific order or ranking. They are used to label distinct entities without implying any hierarchy. In this context, the types of alcoholic drinks such as beer, wine, and hard liquor are examples of nominal variables. They classify the drinks into different groups, but drinking beer isn't necessarily more or less than drinking wine.
  • Nominal variables are non-numeric. Instead, they are often represented using labels, names, or distinct values.
  • These variables are utilized in situations where the importance is on the type or category and not their relative size or order.
  • Examples include gender, religion, geographical location, and types of products.
Nominal variables are fundamental in statistics and categorization, yet they cannot be used in calculations that require numeric inputs, such as finding averages or correlations. This is pivotal in understanding why correlating nominal with ratio variables using a correlation coefficient is not feasible.
Ratio Variables
Unlike nominal variables, ratio variables have a meaningful zero point which allows various mathematical operations. Blood Alcohol Content (BAC) is an example of a ratio variable. It can be measured, compared, and has a clear zero definition鈥攅ssentially the absence of alcohol in the bloodstream.
  • Ratio variables allow for a full range of mathematical computations such as addition, subtraction, multiplication, and division.
  • They enable statistical analyses due to their numeric nature.
  • Other examples include height, weight, and distance.
Ratio variables are integral for conducting analyses in inferential statistics where relationships between variables are examined. However, a key takeaway is that ratio variables like BAC are continuous and numeric, unlike nominal variables.
Correlation Coefficient
The correlation coefficient, often represented by the symbol \(r\), is a statistical measure that indicates the extent to which two continuous variables move in relation to each other. It is designed to capture the strength and direction of a linear relationship between two numerical, continuous variables.
  • The value of \(r\) ranges from -1 to 1.
  • A value of 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation.
  • Correlation only applies to continuous, interval, or ratio-level data.
It is essential to note that correlation coefficients cannot be calculated when one of the variables is nominal. This is due to the absence of a numeric continuum in nominal variables. In the given exercise, trying to correlate a nominal variable (type of drink) with a ratio variable (BAC) does not meet the requirements for computing a proper correlation, rendering any claim of correlation between them statistically unsound.

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Most popular questions from this chapter

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