[DATA] Draw Your Data! Consider the four data sets shown below.
a. Compute the linear correlation coefficient for each data set.

Verified step by step guidance

1. Step 1: Understand the formula for the linear correlation coefficient (Pearson's r), which measures the strength and direction of the linear relationship between two variables x and y. The formula is: \[r = \frac{n \sum xy - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}\] where \(n\) is the number of data points, \(\sum xy\) is the sum of the products of paired scores, \(\sum x\) and \(\sum y\) are the sums of x and y values respectively, and \(\sum x^2\) and \(\sum y^2\) are the sums of the squares of x and y values.
2. Step 2: For each data set, calculate the following sums: - \(\sum x\) (sum of all x values) - \(\sum y\) (sum of all y values) - \(\sum xy\) (sum of the product of each x and y pair) - \(\sum x^2\) (sum of squares of x values) - \(\sum y^2\) (sum of squares of y values) Also, note the number of data points \(n\) in each data set.
3. Step 3: Substitute the calculated sums and \(n\) into the Pearson correlation coefficient formula for each data set separately. This will give you the correlation coefficient \(r\) for each data set.
4. Step 4: Interpret the value of \(r\) for each data set. Recall that \(r\) ranges from -1 to 1, where values close to 1 indicate a strong positive linear relationship, values close to -1 indicate a strong negative linear relationship, and values near 0 indicate little or no linear relationship.
5. Step 5: Summarize the correlation coefficients for all four data sets, comparing their strengths and directions of linear relationships based on the computed \(r\) values.

Key Concepts

Linear Correlation Coefficient

The linear correlation coefficient, often denoted as r, measures the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, where values close to 1 or -1 indicate strong positive or negative linear relationships, respectively, and values near 0 indicate weak or no linear relationship.

Scatterplot and Data Visualization

A scatterplot visually represents paired data points on a coordinate plane, helping to identify patterns, trends, or outliers. Visualizing data sets before calculating correlation can reveal whether a linear model is appropriate or if other relationships exist.

Effect of Outliers on Correlation

Outliers are data points that deviate significantly from the overall pattern. They can disproportionately affect the correlation coefficient, either inflating or deflating its value, thus misleading the interpretation of the relationship between variables.