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Chapter 4: Problem 10

Use the linear correlation coefficient given to determine the coefficient of determination, \(R^{2} .\) Interpret each \(R^{2}\) (a) \(r=-0.32\) (b) \(r=0.13\) (c) \(r=0.40\) (d) \(r=0.93\)

Short Answer

Expert verified
For r=-0.32, R虏=0.1024. For r=0.13, R虏=0.0169. For r=0.40, R虏=0.16. For r=0.93, R虏=0.8649.

Step by step solution

01

- Understanding the Relationship Between r and R虏

The coefficient of determination, denoted as \(R^2\), is calculated by squaring the linear correlation coefficient, denoted as \(r\). It explains the proportion of the variance in the dependent variable that is predictable from the independent variable.
02

- Calculate R虏 for r = -0.32

To find \(R^2\) for \(r = -0.32\), square the value of \(r\): \[R^2 = (-0.32)^2 = 0.1024\] Thus, \(R^2 = 0.1024\). This means approximately 10.24% of the variance in the dependent variable is predictable from the independent variable.
03

- Calculate R虏 for r = 0.13

To find \(R^2\) for \(r = 0.13\), square the value of \(r\): \[R^2 = (0.13)^2 = 0.0169\] Thus, \(R^2 = 0.0169\). This means approximately 1.69% of the variance in the dependent variable is predictable from the independent variable.
04

- Calculate R虏 for r = 0.40

To find \(R^2\) for \(r = 0.40\), square the value of \(r\): \[R^2 = (0.40)^2 = 0.16\] Thus, \(R^2 = 0.16\). This means approximately 16% of the variance in the dependent variable is predictable from the independent variable.
05

- Calculate R虏 for r = 0.93

To find \(R^2\) for \(r = 0.93\), square the value of \(r\): \[R^2 = (0.93)^2 = 0.8649\] Thus, \(R^2 = 0.8649\). This means approximately 86.49% of the variance in the dependent variable is predictable from the independent variable.
06

- Summarize Results

For \(r = -0.32\), \(R^2 = 0.1024\) which means 10.24% predictability. For \(r = 0.13\), \(R^2 = 0.0169\) which means 1.69% predictability. For \(r = 0.40\), \(R^2 = 0.16\) which means 16% predictability. For \(r = 0.93\), \(R^2 = 0.8649\) which means 86.49% predictability.

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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, also known as Pearson's correlation coefficient, is a measure of the strength and direction of a linear relationship between two variables. It's denoted by the symbol \( r \). The value of \( r \) ranges from -1 to 1. When \( r \) is close to 1, it indicates a strong positive linear relationship. Conversely, when \( r \) is close to -1, it shows a strong negative linear relationship. An \( r \) value near 0 suggests no linear correlation between the variables.
variance
Variance is a statistical concept that measures the dispersion or spread of a set of numbers. In simpler terms, it shows how much the numbers in a dataset differ from the mean of the dataset. Higher variance indicates that the numbers are more spread out from the mean, while lower variance indicates that the numbers are close to the mean.
Variance is a crucial concept in calculating the coefficient of determination because \( R^2 \) explains how much of the variance in the dependent variable can be predicted by the independent variable.
predictability
Predictability in statistics refers to the extent to which we can predict the value of one variable based on the value of another variable. The coefficient of determination, \( R^2 \), quantifies this predictability.
For example, if \( R^2 = 0.8649 \), it means 86.49% of the variance in the dependent variable is predictable from the independent variable, thus indicating a high level of predictability. Lower \( R^2 \) values indicate lower predictability. Understanding how much variance is predictable helps analysts make informed decisions based on data-driven insights.

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Most popular questions from this chapter

In Problems \(17-20,\) (a) draw a scatter diagram of the data, (b) by hand, compute the correlation coefficient, and \((c)\) determine whether there is a linear relation between \(x\) and \(y\). $$ \begin{array}{lllrrr} x & 2 & 4 & 6 & 6 & 7 \\ \hline y & 4 & 8 & 10 & 13 & 20 \\ \hline \end{array}$$

(a) By hand, draw a scatter diagram treating \(x\) as the explanatory variable and y as the response variable. (b) Select two points from the scatter diagram and find the equation of the line containing the points selected. (c) Graph the line found in part (b) on the scatter diagram. (d) By hand, determine the least-squares regression line. (e) Graph the least-squares regression line on the scatter diagram. (f) Compute the sum of the squared residuals for the line found in part (b). (g) Compute the sum of the squared residuals for the leastsquares regression line found in part (d). (h) Comment on the fit of the line found in part (b) versus the least-squares regression line found in part ( \(d\) ). $$ \begin{array}{llrlll} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline y & -4 & 0 & 1 & 4 & 5 \\ \hline \end{array} $$

Consider the following set of data: $$ \begin{array}{lllllllll} \hline x & 2.2 & 3.7 & 3.9 & 4.1 & 2.6 & 4.1 & 2.9 & 4.7 \\ \hline y & 3.9 & 4.0 & 1.4 & 2.8 & 1.5 & 3.3 & 3.6 & 4.9 \\ \hline \end{array} $$ (a) Draw a scatter diagram of the data and compute the linear correlation coefficient (b) Draw a scatter diagram of the data and compute the linear correlation coefficient with the additional data point \((10.4,9.3) .\) Comment on the effect the additional data point has on the linear correlation coefficient. Explain why correlations should always be reported with scatter diagrams.

Explain the difference between correlation and causation. When is it appropriate to state that the correlation implies causation?

The wind chill factor depends on wind speed and air temperature. The following data represent the wind speed (in mph) and wind chill factor at an air temperature of \(15^{\circ}\) Fahrenheit. $$ \begin{array}{cc|cc} \begin{array}{l} \text { Wind } \\ \text { Speed, } x \\ (\mathrm{mph}) \end{array} & \begin{array}{l} \text { Wind Chill } \\ \text { Factor, } y \end{array} & \begin{array}{l} \text { Wind } \\ \text { Speed, } x \\ (\mathrm{mph}) \end{array} & \begin{array}{l} \text { Wind Chill } \\ \text { Factor, } y \end{array} \\ \hline 5 & 12 & 25 & -22 \\ \hline 10 & -3 & 30 & -25 \\ \hline 15 & -11 & 35 & -27 \\ \hline 20 & -17 & & \\ \hline \end{array} $$ (a) Draw a scatter diagram of the data treating wind speed as the explanatory variable. (b) Determine the correlation between wind speed and wind chill factor. Does this imply a linear relation between wind speed and wind chill factor? (c) Compute the least-squares regression line. (d) Plot the residuals against the wind speed. (e) Do you think the least-squares regression line is a good model? Why?
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