Questions: Consider the four data sets shown in the accompanying table. Complete parts (a) and (b) below. Click the icon to view the data table. (a) Compute the linear correlation coefficient for each data set. The linear correlation coefficient for the first data set is (Round to three decimal places as needed.) Data Table Full data set Data Set 1 Data Set 2 Data Set 3 Data Set 4 x y x y x y x y 8 9.04 8 10.14 8 8.46 17 7.58 10 7.95 10 9.14 10 7.77 17 6.76 5 8.58 5 9.74 5 13.74 17 8.71 9 9.81 9 9.77 9 8.11 17 9.84 7 9.33 7 10.26 7 8.81 17 9.47 4 10.96 4 9.10 4 9.84 17 8.04 12 8.24 12 7.13 12 7.08 17 6.25 14 5.26 14 4.10 14 6.39 17 6.56 6 11.84 6 10.13 6 9.15 17 8.91 11 5.82 11 8.26 11 7.42 17 7.89 13 6.68 13 5.47 13 6.73 6 13.50

Consider the four data sets shown in the accompanying table. Complete parts (a) and (b) below. Click the icon to view the data table. (a) Compute the linear correlation coefficient for each data set. The linear correlation coefficient for the first data set is (Round to three decimal places as needed.) Data Table Full data set Data Set 1 Data Set 2 Data Set 3 Data Set 4 x y x y x y x y 8 9.04 8 10.14 8 8.46 17 7.58 10 7.95 10 9.14 10 7.77 17 6.76 5 8.58 5 9.74 5 13.74 17 8.71 9 9.81 9 9.77 9 8.11 17 9.84 7 9.33 7 10.26 7 8.81 17 9.47 4 10.96 4 9.10 4 9.84 17 8.04 12 8.24 12 7.13 12 7.08 17 6.25 14 5.26 14 4.10 14 6.39 17 6.56 6 11.84 6 10.13 6 9.15 17 8.91 11 5.82 11 8.26 11 7.42 17 7.89 13 6.68 13 5.47 13 6.73 6 13.50
Transcript text: Consider the four data sets shown in the accompanying table. Complete parts (a) and (b) below. Click the icon to view the data table. (a) Compute the linear correlation coefficient for each data set. The linear correlation coefficient for the first data set is (Round to three decimal places as needed.) Data Table Full data set \begin{tabular}{cccccccc} \multicolumn{2}{c}{ Data Set $\mathbf{1}$} & \multicolumn{2}{c}{ Data Set $\mathbf{2}$} & \multicolumn{2}{c}{ Data Set $\mathbf{3}$} & \multicolumn{2}{c}{ Data Set 4 } \\ $\mathbf{x}$ & $\mathbf{y}$ & $\mathbf{x}$ & $\mathbf{y}$ & $\mathbf{x}$ & $\mathbf{y}$ & $\mathbf{x}$ & $\mathbf{y}$ \\ 8 & 9.04 & 8 & 10.14 & 8 & 8.46 & 17 & 7.58 \\ 10 & 7.95 & 10 & 9.14 & 10 & 7.77 & 17 & 6.76 \\ 5 & 8.58 & 5 & 9.74 & 5 & 13.74 & 17 & 8.71 \\ 9 & 9.81 & 9 & 9.77 & 9 & 8.11 & 17 & 9.84 \\ 7 & 9.33 & 7 & 10.26 & 7 & 8.81 & 17 & 9.47 \\ 4 & 10.96 & 4 & 9.10 & 4 & 9.84 & 17 & 8.04 \\ 12 & 8.24 & 12 & 7.13 & 12 & 7.08 & 17 & 6.25 \\ 14 & 5.26 & 14 & 4.10 & 14 & 6.39 & 17 & 6.56 \\ 6 & 11.84 & 6 & 10.13 & 6 & 9.15 & 17 & 8.91 \\ 11 & 5.82 & 11 & 8.26 & 11 & 7.42 & 17 & 7.89 \\ 13 & 6.68 & 13 & 5.47 & 13 & 6.73 & 6 & 13.50 \end{tabular}
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Solution

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Solution Steps

Solution Approach

To compute the linear correlation coefficient for each data set, we will use the Pearson correlation formula. This involves calculating the covariance of the two variables and dividing it by the product of their standard deviations. We will use Python's numpy library to handle these calculations efficiently.

Step 1: Understanding the Linear Correlation Coefficient

The linear correlation coefficient, denoted as \( r \), measures the strength and direction of a linear relationship between two variables. It is calculated using the formula:

\[ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} \]

where \( x_i \) and \( y_i \) are the data points, and \( \bar{x} \) and \( \bar{y} \) are the means of the \( x \) and \( y \) data sets, respectively.

Step 2: Calculating the Correlation Coefficient for Each Data Set

Using the formula above, we calculate the correlation coefficient for each data set. The results are as follows:

  • For Data Set 1: \( r_1 = -0.816 \)
  • For Data Set 2: \( r_2 = -0.817 \)
  • For Data Set 3: \( r_3 = -0.816 \)
  • For Data Set 4: \( r_4 = -0.817 \)
Step 3: Interpreting the Results

The correlation coefficients for all data sets are negative, indicating an inverse relationship between the variables in each data set. The values are close to \(-1\), suggesting a strong negative linear relationship.

Final Answer

The linear correlation coefficients for the data sets are:

  • Data Set 1: \(\boxed{-0.816}\)
  • Data Set 2: \(\boxed{-0.817}\)
  • Data Set 3: \(\boxed{-0.816}\)
  • Data Set 4: \(\boxed{-0.817}\)
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