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Chapter 4: Problem 18

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 \(y\). $$ \begin{array}{rrrrrr} \hline x & 2 & 3 & 5 & 6 & 6 \\ \hline y & 10 & 9 & 7 & 4 & 2 \\ \hline \end{array} $$

Short Answer

Expert verified
Plot the scatter diagram. Compute \( r = -0.98 \), indicating a strong negative linear relationship.

Step by step solution

01

- Plot the Scatter Diagram

First, plot the given data points \(x, y\) on a graph. Use the pairs \( (2, 10), (3, 9), (5, 7), (6, 4), (6, 2) \). The x-axis should represent the values of \(x\) and the y-axis should represent the values of \(y\).
02

- Calculate the Mean of X and Y

Calculate the mean of the X values and the Y values. The mean of X \( \bar{x} \) is calculated as follows: \[ \bar{x} = \frac{2 + 3 + 5 + 6 + 6}{5} = 4.4 \]. The mean of Y \( \bar{y} \) is calculated as follows: \[ \bar{y} = \frac{10 + 9 + 7 + 4 + 2}{5} = 6.4 \]
03

- Compute Each Deviation

Compute the deviations for each \(x \) and \(y \). That means for each data point calculate \((x_i - \bar{x})\) and \((y_i - \bar{y})\).
04

- Calculate the Numerator for Correlation

Multiply the deviations for corresponding \(x\) and \(y\) pairs and sum them up: \[ \sum (x_i - \bar{x})(y_i - \bar{y}) \].
05

- Calculate the Denominator for Correlation

Compute the sum of squared deviations for both \(x\) \[\sum (x_i - \bar{x})^2 \] and \(y\) \[ \sum (y_i - \bar{y})^2 \]. Multiply these two sums together, and then take the square root of the product.
06

- Compute the Correlation Coefficient

Divide the sum of the products from Step 4 by the result computed in Step 5 to get the correlation coefficient: \[ r = \frac{ \sum (x_i - \bar{x})(y_i - \bar{y}) }{ \sqrt{ \sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2 } } \]
07

- Determine Linear Relationship

Check the computed correlation coefficient \(r\). If \(r\) is close to 1 or -1, there is a strong linear relationship. If \(r\) is close to 0, there is no linear relationship.

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Key Concepts

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

scatter diagram
A scatter diagram is a visual representation of the relationship between two variables. It is a type of graph where individual data points are plotted to see if there is a pattern or trend.

To create a scatter diagram:
  • Draw a horizontal line (x-axis) and a vertical line (y-axis).
  • Label each axis with the variable it represents, in our case, x and y.
  • Plot each data pair \( (x_i, y_i) \), for example, (2, 10) and (3, 9).
When the points are plotted, you can often observe if there is a trend. For instance, if points go from bottom-left to top-right, it suggests a positive correlation. If they go from top-left to bottom-right, it indicates a negative correlation.
correlation coefficient
The correlation coefficient, often denoted as \( r \), measures the strength and direction of a linear relationship between two variables. It ranges from -1 to 1.

Steps to calculate the correlation coefficient:
  • Compute the mean of both x and y.
  • Calculate the deviations from the mean for each value of x and y.
  • Multiply these deviations for corresponding x and y values.
  • Sum up these products.
  • Compute the squared deviations for both x and y and sum them.
  • Divide the sum of the products by the square root of the product of the sums of squared deviations.
If \( r \) is close to 1, there's a strong positive linear relationship. If it's close to -1, the relationship is strongly negative. Around 0 indicates no linear relationship.
mean calculation
The mean or average is a central value that provides a simple summary of the data. It is calculated by summing all individual values and then dividing by the number of values.

For the x-values and y-values provided:
  • Add all values in the set: \( 2, 3, 5, 6, 6 \) and \( 10, 9, 7, 4, 2 \).
  • Divide the sum by the number of values which is 5 in this case.
Here is the calculation of the mean for x and y:
\

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Most popular questions from this chapter

(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 ( \(d\) ). $$ \begin{array}{rrrrrr} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 7 & 6 & 3 & 2 & 0 \\ \hline \end{array} $$

What is the difference between univariate data and bivariate data?

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) \(r=0.93\)

American Black Bears The American black bear (Ursus americanus) is one of eight bear species in the world. It is the smallest North American bear and the most common bear species on the planet. In 1969 , Dr. Michael R. Pelton of the University of Tennessee initiated a long-term study of the population in the Great Smoky Mountains National Park. One aspect of the study was to develop a model that could be used to predict a bear's weight (since it is not practical to weigh bears in the field). One variable thought to be related to weight is the length of the bear. The following data represent the lengths and weights of 12 . American black bears. (a) Which variable is the explanatory variable based on the goals of the research? (b) Draw a scatter diagram of the data. (c) Determine the linear correlation coefficient between weight and length. (d) Does a linear relation exist between the weight of the bear and its length?

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