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Chapter 3: Problem 17

Correlation blunders Fach of the following statements contains an error. Explain what's wrong in each case. (a) There is a high correlation between the gender of American workers and their income." (b) "We found a high correlation \((r=1.09)\) between students' ratings of faculty teaching and ratings made by other faculty members." (c) The correlation between planting rate and yield of corn was found to be \(r=0.23\) bushel."

Short Answer

Expert verified
(a) Cannot correlate categorical and quantitative variables; (b) Correlation cannot exceed 1; (c) Correlation is dimensionless.

Step by step solution

01

Understanding Correlation Basics

Before exploring the errors in the statements, we must understand that correlation measures the strength and direction of a linear relationship between two quantitative variables. Correlation coefficients range from -1 to 1, where 1 means a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 signifies no linear relationship.
02

Analyzing Statement (a)

The statement 'There is a high correlation between the gender of American workers and their income' is incorrect because correlation requires both variables to be quantitative. Gender is a categorical variable (e.g., male or female), not quantitative. Therefore, it is inappropriate to measure correlation between a categorical and a quantitative variable.
03

Analyzing Statement (b)

The statement 'We found a high correlation \((r=1.09)\) between students' ratings of faculty teaching and ratings made by other faculty members' contains an error in the correlation coefficient value. Correlation coefficients cannot exceed 1; therefore, a correlation value of \(r=1.09\) is impossible.
04

Analyzing Statement (c)

The statement 'The correlation between planting rate and yield of corn was found to be \(r=0.23\) bushel' is incorrect because correlation coefficients are dimensionless and should not have units. The phrase '0.23 bushel' is incorrect as the correlation should simply be stated as \(r=0.23\).

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Key Concepts

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

Correlation Coefficient
The correlation coefficient is a numerical value that measures the strength and direction of a linear relationship between two quantitative variables. This coefficient is denoted by the letter \( r \) and ranges from -1 to 1. Here are key points about its characteristics:
  • A correlation of \( r = 1 \) signifies a perfect positive linear relationship, meaning as one variable increases, the other increases proportionally.
  • A correlation of \( r = -1 \) indicates a perfect negative linear relationship, meaning as one variable increases, the other decreases proportionally.
  • A correlation of \( r = 0 \) suggests there is no linear relationship between the variables.
It's important to remember that correlation does not imply causation. Even if two variables are correlated, it doesn鈥檛 mean that changes in one variable directly cause changes in the other.
Quantitative Variables
Quantitative variables are variables that represent measurable quantities and can be expressed numerically. They allow us to perform a wide range of mathematical operations, such as addition, subtraction, and calculating averages. Examples include height, weight, income, and age.
Quantitative variables are essential in correlation calculations, as correlation requires assessing the linear relationship between two such variables. When dealing with non-quantitative, or categorical, variables, correlation is not an appropriate measure.
Categorical Variables
Categorical variables represent characteristics or qualities that can be sorted into categories rather than measured numerically. Examples include gender, type of climate, and brand preference. Unlike quantitative variables, categorical variables are not suitable for correlation calculations because they do not have inherent numerical values or order. Attempting to calculate correlation between a quantitative and categorical variable, as seen in Statement (a) of the exercise, is a common error. Instead, other statistical methods such as chi-square tests may be used to explore relationships with categorical data.
Linear Relationship
A linear relationship between two variables occurs when their association can be represented on a graph as a straight line. In such scenarios, changes in one variable are directly proportional to changes in the other. Key characteristics of linear relationships include:
  • They can be positive, where both variables increase together, or negative, where one variable decreases as the other increases.
  • Linear relationships are fundamental for correlation analysis, as correlation specifically measures the degree of linearity.
If data points form a pattern that is not straight, the relationship could be non-linear or could require transformation to become linear. Understanding the nature of the relationship is vital for accurate analysis and interpretation.

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Most popular questions from this chapter

Acid rain Researchers studying acid rain measured the acidity of precipitation in a Colorado wilderness area for 150 consecutive weeks. Acidity is measured by pH. Lower pH values show higher acidity. The researchers observed a linear pattern over time. They reported that the regression line \(\mathrm{pH}=5.43-\) 0.0053 (weeks) fit the data well. (a) Identify the slope of the line and explain what itmeans in this setting. (b) Identify the \(y\) intercept of the line and explain what it means in this setting. (c) According to the regression line, what was the pH at the end of this study?

What's my line? You use the same bar of soap to shower each morning. The bar weighs 80 grams when it is new. Its weight goes down by 6 grams per day on the average. What is the equation of the regression line for predicting weight from days of use?

Hot dogs Are hot dogs that are high in calories also high in salt? The figure below is a scatterplot of the calories and salt content (measured as milligrams of sodium) in 17 brands of meat hot dogs. (a) The correlation for these data is r = 0.87. Explainwhat this value means. (b) What effect would removing the hot dog brand with the lowest calorie content have on the correlation? Justify your answer.

Beer and blood alcohol The example on page 182 describes a study in which adults drank different amounts of beer. The response variable was their blood alcohol content (BAC). BAC for the same amount of beer might depend on other facts about the subjects. Name two other variables that could ac- count for the fact that \(r^{2}=0.80\) .

Student loans \((2.2)\) A government report looked at the amount borrowed for college by students who graduated in 2000 and had taken out student loans. 12 The mean amount was \(\overline{x}=\$ 17,776\) and the standard deviation was \(s_{x}=\$ 12,034\) . The median was \(\$ 15,532\) and the quartiles were \(Q_{1}=\$ 9900\) and \(Q_{3}=\$ 22,500\) (a) Compare the mean and the median. Also compare the distances of \(Q_{1}\) and \(Q_{3}\) from the median. Explain why both comparisons suggest that the distribution is right-skewed. (b) The right-skew pulls the standard deviation up. So a Normal distribution with the same mean and standard deviation would have a third quartile larger than the actual \(Q_{3}\) . Find the third quartile of the Normal distribution with \(\mu=\$ 17,776\) and \(\sigma=\$ 12,034\) and compare it with \(Q_{3}=\$ 22,500\) .
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