Question

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).

Short answer

a. The regression equation is\(\hat y = 0.085 - 0.985x\).

b. The regression equation with only 4 lower-left corner values is\(\hat y = 1.5 - 0.00x\).

c. The regression equation with only 4 upper-right corner values is\(\hat y = 9.5 - 0.00x\).

c. The regression equations obtained in parts (a), (b), and (c) are completely different from one another. The presence of different sets of values affects the regression equation to a large extent.

Step-by-step solution

Given information

A set of 8 pairs of values is considered.

Regression equation using all values

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.

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

Regression equation using only lower-left points

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

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

Regression equation using upper-right corner values

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

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

Comparison

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.