--- ## Analysis of Bivariate Data Sets ### Introduction Below are four bivariate data sets alongside the scatter plot for each. Each scatter plot is displayed on the same scale, representing sample values drawn from a population. The details of each data set and its corresponding scatter plot are elaborated on below. ### Data Sets and Scatter Plots #### Figure 1 - **Data Set \(x, y\):** ``` x y 10 8.4 9.0 6.6 8.0 6.8 7.0 4.3 6.0 3.0 5.0 4.2 4.0 7.5 3.0 6.3 2.0 6.2 1.0 2.7 ``` - **Scatter Plot Description:** This scatter plot visualizes the data set \(x\) versus \(y\). The points appear to be distributed in a pattern where \(y\) starts high when \(x\) is high, decreases as \(x\) decreases but shows a point where it suddenly starts increasing again. #### Figure 2 - **Data Set \(u, v\):** ``` u v 10 47 9.0 42 8.0 39 7.0 50 6.0 68 5.0 87 4.0 84 3.0 75 2.0 75 1.0 77 ``` - **Scatter Plot Description:** The scatter plot corresponds to the data set \(u\) versus \(v\). The points indicate a sharp increase in \(v\) values as \(u\) decreases, showcasing a non-linear relationship that might imply an exponential growth or some outliers. #### Figure 3 - **Data Set \(w, t\):** ``` w t 10 78 9.0 91 8.0 87 7.0 45 ### Correlation Analysis Exercise **Instructions:** Answer the following questions about the relationships between pairs of variables and the values of \( r \), the sample correlation coefficient. The same response may be the correct answer for more than one question. **Questions and Selection Options:** 1. **For which data set is the correlation coefficient \( r \) equal to 1?** - **Choose One** (Dropdown menu for selection) 2. **Which data set indicates the strongest positive linear relationship between its two variables?** - **Choose One** (Dropdown menu for selection) 3. **For which data set is the correlation coefficient \( r \) closest to 0?** - **Choose One** (Dropdown menu for selection) 4. **For which data set is the correlation coefficient \( r \) closest to -1?** - **Choose One** (Dropdown menu for selection) **Instructions for Selection:** - Use the dropdown menus to choose the appropriate data set for each question based on the sample correlation coefficient values. **Explanation of Correlation Coefficients:** - The correlation coefficient \( r \) measures the strength and direction of a linear relationship between two variables. - \( r = 1 \) indicates a perfect positive linear relationship. - \( r = -1 \) indicates a perfect negative linear relationship. - \( r = 0 \) indicates no linear relationship. - Values between -1 and 1 indicate the degree of linear correlation, with values closer to 1 or -1 indicating stronger correlations. **Note:** Ensure that you analyze the provided data sets thoroughly to determine the correct correlation coefficient for each set. This will involve evaluating the given data points and calculating the correlation coefficient \( r \) where necessary. This exercise aims to reinforce understanding of correlation coefficients and their interpretation in statistical analysis.