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Chapter 8: Problem 7

The correlation between a cereal's fiber and potassium contents is \(r=0.903\). What fraction of the variability in potassium is accounted for by the amount of fiber that servings contain?

Short Answer

Expert verified
81.54% of the variability in potassium is accounted for by fiber.

Step by step solution

01

Understand the Correlation Coefficient

The correlation coefficient, denoted as \( r \), is a measure of the strength and direction of a linear relationship between two variables. In this case, \( r = 0.903 \) indicates a strong positive correlation between fiber and potassium content in cereal.
02

Calculate the Coefficient of Determination

The fraction of variability in one variable (potassium) that can be predicted from another variable (fiber) is given by the coefficient of determination, \( r^2 \). Compute \( r^2 \) by squaring the correlation coefficient: \( r^2 = (0.903)^2 \).
03

Perform the Square Calculation

Calculate \( r^2 = 0.903^2 = 0.815409 \). This calculation tells us that approximately 81.54% of the variability in potassium content is accounted for by the fiber content.
04

Interpret the Results

The coefficient of determination, \( r^2 = 0.815 \), suggests that about 81.54% of the variability in the potassium content of cereals can be explained by the amount of fiber they contain. This strong relationship implies that fiber content is a good predictor of potassium content in cereals.

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Key Concepts

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

Coefficient of Determination
The coefficient of determination is a key concept in statistics when assessing the predictive power of a relationship between two variables, such as fiber and potassium content in cereal. It is symbolized as \( r^2 \), where \( r \) is the correlation coefficient.
  • The coefficient of determination is calculated by squaring the correlation coefficient \( r \).
  • It provides the proportion of the variance in the dependent variable that is predictable from the independent variable.
  • This value, expressed as a percentage, indicates how much of the variability in the outcome variable (in this case, potassium content) can be explained by the predictor variable (fiber content).
For instance, in this exercise, the coefficient of determination \( r^2 \) is calculated as \( (0.903)^2 = 0.815409 \), which means that approximately 81.54% of the variability in potassium content is accounted for by fiber content. The higher the \( r^2 \), the better the predictive power of the model.
Variability
Variability refers to how spread out or dispersed the values of a dataset are. Greater variability means more differences among data points.
  • In the context of this exercise, it relates to potassium content among different cereals.
  • Understanding variability is essential because it gives insights into the reliability and consistency of the data.
  • If one variable can explain a significant portion of the variability in another (like fiber explaining potassium content), it indicates a strong predictive relationship.
Given that approximately 81.54% of the variability in potassium content is explained by fiber content, it suggests that this relationship is quite strong. Lower unexplained variability means the model is a good fit for the data.
Linear Relationship
A linear relationship is one of the fundamental forms of relationships in statistics. It occurs when two variables increase or decrease at a consistent rate, forming a straight line when plotted on a graph.
  • The correlation coefficient \( r \) quantifies how closely the data fits a linear pattern.
  • An \( r \) value near 1 or -1 indicates a very strong linear relationship, while a value near 0 suggests a weak one.
  • In this context, \( r = 0.903 \) represents a strong positive linear relationship. As fiber content increases, potassium content tends to increase as well.
Understanding linear relationships is crucial for predicting the behavior of one variable based on another, particularly when planning experiments or making data-driven decisions.

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Most popular questions from this chapter

A Biology student who created a regression model to use a bird's Height when perched for predicting its Wingspan made these two statements. Assuming the calculations were done correctly, explain what is wrong with each interpretation. a) \(\mathrm{My} R^{2}\) of \(93 \%\) shows that this linear model is appropriate. b) A bird 10 inches tall will have a wingspan of 17 inches.

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