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Chapter 10: Problem 7

Use the value of the linear correlation coefficient \(r\) to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables. \(r=0.885(x=\text { weight of male, } y=\text { waist size of male })\)

Short Answer

Expert verified
The coefficient of determination is 0.783, which means that 78.3\text{\textpercent} of the total variation can be explained by the linear relationship between the two variables.

Step by step solution

01

Understanding the linear correlation coefficient

The linear correlation coefficient, denoted as \(r\), measures the strength and direction of a linear relationship between two variables. For this problem, \(r=0.885\). The value of \(r\) ranges from -1 to 1.
02

Calculating the coefficient of determination

To find the coefficient of determination, denoted as \(r^2\), square the value of the linear correlation coefficient. Therefore, \(r^2 = (0.885)^2\).
03

Calculating the value of the coefficient of determination

Perform the calculation: \(r^2 = 0.885 \times 0.885 = 0.783225\). So, the coefficient of determination \(r^2\) is 0.783.
04

Converting the coefficient of determination to a percentage

To find the percentage of the total variation that can be explained by the linear relationship between the two variables, multiply the coefficient of determination by 100. Therefore, \(0.783 \times 100 = 78.3\text{\textpercent} \).

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Key Concepts

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

linear correlation coefficient
The linear correlation coefficient, denoted by the symbol \( r \), quantifies the strength and direction of a linear relationship between two variables. In this exercise, the linear correlation coefficient \( r \) is given as 0.885. Values of \( r \) can range from -1 to 1.

A value close to 1 or -1 indicates a strong linear relationship, while values near 0 suggest a weak or no linear relationship.

For instance, if \( r \) is positive, as in our case, it means that as one variable increases, the other one also increases. If \( r \) were negative, it would mean that as one variable increases, the other one decreases. This is critical in understanding how closely two variables, such as weight and waist size in our problem, move together in a linear fashion.
percentage of total variation
The percentage of total variation explained by the linear relationship between two variables is derived from the coefficient of determination, denoted as \( r^2 \). In our exercise, we obtained \( r^2 \) by squaring \( r = 0.885 \), giving us 0.783.

To convert this into a percentage, we multiply by 100, resulting in 78.3\%. This percentage quantifies the proportion of the variation in the dependent variable (waist size) that can be attributed to its linear relationship with the independent variable (weight).

This means that 78.3\% of the total variation in men's waist size can be explained by their weight. The remaining 21.7\% of the variation is due to other factors or random noise not accounted for by this linear relationship.
linear relationship between variables
A linear relationship between two variables indicates that a change in one variable is associated with a proportional change in the other variable. This is often visualized using a straight line on a scatter plot.

In our exercise, the linear relationship between weight and waist size is quite strong, as indicated by the high linear correlation coefficient (\( r \) = 0.885). This implies that we can predict the waist size of a male with reasonable accuracy if we know his weight.

Such a relationship is useful in many fields, such as health and fitness, where understanding the relationship between different body measurements can help in assessing health risks and tailoring fitness programs. By knowing this linear relationship, we can make better-informed decisions based on statistical evidence.

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Most popular questions from this chapter

Find the (a) explained variation, (b) unexplained variation, and (c) indicated prediction interval. In each case, there is sufficient evidence to support a claim of a linear correlation, so it is reasonable to use the regression equation when making predictions. The table below lists overhead widths (cm) of seals measured from photographs and the weights (kg) of the seals (based on "Mass Estimation of Weddell Seals Using Techniques of Photogrammetry," by R. Garrott of Montana State University). For the prediction interval, use a \(99 \%\) confidence level with an overhead width of \(9.0 \mathrm{cm} .\) $$\begin{array}{l|l|l|l|l|l|l}\hline \text { Overhead Width } & 7.2 & 7.4 & 9.8 & 9.4 & 8.8 & 8.4 \\ \hline \text { Weight } & 116 & 154 & 245 & 202 & 200 & 191 \\\\\hline\end{array}$$

Use the value of the linear correlation coefficient \(r\) to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables. \(r=0.783(x=\text { head width of a bear, } y=\text { weight of a bear })\)

Let \(x\) represent years coded as \(1,2,3, \ldots\). for years starting in 1980 and let \(y\) represent the numbers of points scored in each Super Bowl from \(1980 .\) Using the data from 1980 to the last Super Bowl at the time of this writing, we obtain the following values of \(R^{2}\) for the different models: linear: 0.147; quadratic: 0.255; logarithmic: 0.176; exponential: 0.175; power: \(0.203 .\) Based on these results, which model is best? Is the best model a good model? What do the results suggest about predicting the number of points scored in a future Super Bowl game?

What is the relationship between the linear correlation coefficient \(r\) and the slope \(b_{1}\) of a regression line?

Construct a scatterplot, and find the value of the linear correlation coefficient \(r\) Also find the \(P\) -value or the critical values of \(r\) from Table \(A\) -6. Use a significance level of \(\alpha=0.05 .\) Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section \(10-2\) exercises.)Listed below are numbers of registered pleasure boats in Florida (tens of thousands) and the numbers of manatee fatalities from encounters with boats in Florida for each of several recent years. The values are from Data Set 10 "Manatee Deaths" in Appendix B. Is there sufficient evidence to conclude that there is a linear correlation between numbers of registered pleasure boats and numbers of manatee boat fatalities?$$\begin{array}{|l|l|l|l|l|l|l|l|l|l|}\hline \text { Pleasure Boats } & 99 & 99 & 97 & 95 & 90 & 90 & 87 & 90 & 90 \\ \hline \text { Manatee Fatalities } & 92 & 73 & 90 & 97 & 83 & 88 & 81 & 73 & 68 \\ \hline\end{array}$$
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