In Problems 17–20, (b) by hand, compute the correlation coefficient, and (c) determine whether there is a linear relation between x and y.

Verified step by step guidance

1. Step 1: Organize the data points given as pairs \((x, y)\): \((2, 10), (3, 9), (5, 7), (6, 4), (6, 2)\).
2. Step 2: Calculate the means of \(x\) and \(y\) using the formulas: \[\bar{x} = \frac{\sum x_i}{n}\] \[\bar{y} = \frac{\sum y_i}{n}\] where \(n\) is the number of data points.
3. Step 3: Compute the deviations from the mean for each \(x_i\) and \(y_i\), then calculate the products of these deviations, the squared deviations for \(x\), and the squared deviations for \(y\). Specifically, find: \[\sum (x_i - \bar{x})(y_i - \bar{y}), \quad \sum (x_i - \bar{x})^2, \quad \sum (y_i - \bar{y})^2\]
4. Step 4: Use the formula for the Pearson correlation coefficient \(r\): \[r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}\] This formula measures the strength and direction of the linear relationship between \(x\) and \(y\).
5. Step 5: Interpret the value of \(r\): - If \(r\) is close to 1 or -1, there is a strong linear relationship. - If \(r\) is close to 0, there is little to no linear relationship. Use this to determine whether a linear relation exists between \(x\) and \(y\).

Key Concepts

Correlation Coefficient

The correlation coefficient measures the strength and direction of a linear relationship between two variables, x and y. 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 suggest little or no linear correlation.

Calculation of Correlation Coefficient by Hand

To compute the correlation coefficient by hand, use the formula involving the covariance of x and y divided by the product of their standard deviations. This requires calculating means, deviations from the means, and sums of squares for both variables, then applying these values to the formula.

Determining Linear Relationship

Determining if a linear relationship exists involves interpreting the correlation coefficient and examining the scatterplot of paired data. A strong correlation coefficient (close to ±1) and a pattern of points roughly forming a straight line indicate a linear relationship between x and y.