Effects of Clusters Refer to the Minitab-generated scatterplot given in Exercise 12 of Section 10-1 on page 485.

a. Using the pairs of values for all 8 points, find the equation of the regression line.

b. Using only the pairs of values for the 4 points in the lower left corner, find the equation of the regression line.

c. Using only the pairs of values for the 4 points in the upper right corner, find the equation of the regression line.

d. Compare the results from parts (a), (b), and (c).

A set of 8 pairs of values is considered.

a.

The regression equation of y on x has the following notation:

\(\hat y = {b_0} + {b_1}x\),where

\({b_0}\)is the intercept term, and

\({b_1}\) is the slope coefficient.

The following data points are considered:

The following table shows the necessary calculations:

The value of the y-intercept is computed below.

\(\begin{array}{c}{b_0} = \frac{{\left( {\sum y } \right)\left( {\sum {{x^2}} } \right) - \left( {\sum x } \right)\left( {\sum {xy} } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ = \frac{{\left( {44} \right)\left( {372} \right) - \left( {44} \right)\left( {370} \right)}}{{8\left( {372} \right) - {{\left( {44} \right)}^2}}}\\ = 0.085\end{array}\).

The value of the slope coefficient is computed below.

\(\begin{array}{c}{b_1} = \frac{{n\left( {\sum {xy} } \right) - \left( {\sum x } \right)\left( {\sum y } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ = \frac{{\left( 8 \right)\left( {370} \right) - \left( {44} \right)\left( {44} \right)}}{{8\left( {372} \right) - {{\left( {44} \right)}^2}}}\\ = 0.985\end{array}\).

b.

The following 4 pairs of data points are considered:

The following table shows the necessary calculations:

The value of the y-intercept is computed below.

\(\begin{array}{c}{b_0} = \frac{{\left( {\sum y } \right)\left( {\sum {{x^2}} } \right) - \left( {\sum x } \right)\left( {\sum {xy} } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ = \frac{{\left( 6 \right)\left( {10} \right) - \left( 6 \right)\left( 9 \right)}}{{4\left( {10} \right) - {{\left( 6 \right)}^2}}}\\ = 1.5\end{array}\).

The value of the slope coefficient is computed below.

\(\begin{array}{c}{b_1} = \frac{{n\left( {\sum {xy} } \right) - \left( {\sum x } \right)\left( {\sum y } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ = \frac{{\left( 4 \right)\left( 9 \right) - \left( 6 \right)\left( 6 \right)}}{{4\left( {10} \right) - {{\left( 6 \right)}^2}}}\\ = 0.000\end{array}\).

Thus, the regression equation becomes

\(\hat y = 1.5 - 0.00x\).

c.

The following 4 pairs of data points are considered:

The following table shows the necessary calculations:

The value of the y-intercept is computed below.

\(\begin{array}{c}{b_0} = \frac{{\left( {\sum y } \right)\left( {\sum {{x^2}} } \right) - \left( {\sum x } \right)\left( {\sum {xy} } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ = \frac{{\left( {38} \right)\left( {362} \right) - \left( {38} \right)\left( {361} \right)}}{{4\left( {362} \right) - {{\left( {38} \right)}^2}}}\\ = 9.5\end{array}\).

The value of the slope coefficient is computed below.

\(\begin{array}{c}{b_1} = \frac{{n\left( {\sum {xy} } \right) - \left( {\sum x } \right)\left( {\sum y } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ = \frac{{\left( 4 \right)\left( {361} \right) - \left( {38} \right)\left( {38} \right)}}{{4\left( {362} \right) - {{\left( {38} \right)}^2}}}\\ = 0.000\end{array}\).

Thus, the regression equation becomes

\(\hat y = 9.5 - 0.00x\).

d.

The regression equations obtained in parts (a), (b), and (c) are completely different from one another.

Thus, the presence of different sets of values can greatly influence the regression equation.