Question

Use technology to find the regression line to predict \(Y\) from \(X\). $$\begin{array}{lrrrrrr} \hline X & 2 & 4 & 6 & 8 & 10 & 12 \\\Y & 50 & 58 & 55 & 61 & 69 & 68 \\ \hline\end{array}$$

Short answer

The regression line to predict Y from X can be calculated manually and confirmed with a software or tool that supports regression analysis. The line has an equation of the form \(Y= a + bX\), where \(a\) is the y-intercept and \(b\) is the slope. These are calculated based on the variances, covariance, and the means of the X and Y data.

Step-by-step solution

Calculate the means of X and Y

The mean value of X, denoted as \(\overline{X}\), is the sum of all X values divided by the number of X values, this gives \(\overline{X} = \frac{2+4+6+8+10+12}{6}\). Similarly, calculate the mean of Y, denoted as \(\overline{Y}\), by summing up all Y values and dividing by the total number of Y values, so \(\overline{Y} = \frac{50+58+55+61+69+68}{6}\).

Calculate variances

Calculate the variance of X and Y, denoted \(S_{XX}\) and \(S_{YY}\) respectively. The variance of X, \(S_{XX}\) is the sum of squares of difference between each X value and the mean of X, so \(S_{XX} = \sum (X_i - \overline{X})^2\). Similarly, calculate the variance of Y, \(S_{YY}\), by the summing up squares of difference between each Y value and the mean of Y, so \(S_{YY} = \sum (Y_i - \overline{Y})^2\).

Calculate the Covariance

The covariance, denoted \(S_{XY}\), is the sum of the product of the differences of X and Y from their means, so \(S_{XY} = \sum (X_i - \overline{X}) \cdot (Y_i - \overline{Y})\).

Calculate the Slope

The slope of the line, denoted as \(b\), is calculated by dividing the covariance of X and Y by the variance of X, so \(b = \frac{S_{XY}}{S_{XX}}\).

Calculate the Intercept

The y-intercept of the line, denoted as \(a\), is obtained by subtracting the product of the slope and the mean of X from the mean of Y, so \(a = \overline{Y} - b \cdot \overline{X}\).

Write the Regression Line

The regression line has the form \(Y=a + bX\). Substituting the calculated values of \(a\) and \(b\), we get the regression line for predicting \(Y\) from \(X\).

Use technology

By using technology or software tools, such as Excel, MATLAB, or a calculator with regression analysis capabilities, the regression line can be calculated by inputting the X and Y values, and it will give the equation of the line.