91影视

A set of 10 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.

\(\begin{array}{c}{b_0} = \frac{{\left( {\sum y } \right)\left( {\sum {{x^2}} } \right) - \left( {\sum x } \right)\left( {\sum {xy} } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ = \frac{{\left( {28} \right)\left( {142} \right) - \left( {28} \right)\left( {136} \right)}}{{10\left( {142} \right) - {{\left( {28} \right)}^2}}}\\ = 0.264\end{array}\).

The value of the slope coefficient is computed below.

\(\begin{array}{c}{b_1} = \frac{{n\left( {\sum {xy} } \right) - \left( {\sum x } \right)\left( {\sum y } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ = \frac{{\left( {10} \right)\left( {136} \right) - \left( {28} \right)\left( {28} \right)}}{{10\left( {142} \right) - {{\left( {28} \right)}^2}}}\\ = 0.906\end{array}\).

Thus, the regression equation becomes

\(\hat y = 0.264 - 0.906x\).

b.

The following 9 pairs of data points are considered:

The following table shows the necessary calculations:

The value of the y-intercept is computed below.

\(\begin{array}{c}{b_0} = \frac{{\left( {\sum y } \right)\left( {\sum {{x^2}} } \right) - \left( {\sum x } \right)\left( {\sum {xy} } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ = \frac{{\left( {18} \right)\left( {42} \right) - \left( {18} \right)\left( {36} \right)}}{{9\left( {42} \right) - {{\left( {18} \right)}^2}}}\\ = 2.000\end{array}\).

The value of the slope coefficient is computed below.

\(\begin{array}{c}{b_1} = \frac{{n\left( {\sum {xy} } \right) - \left( {\sum x } \right)\left( {\sum y } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ = \frac{{\left( 9 \right)\left( {36} \right) - \left( {18} \right)\left( {18} \right)}}{{9\left( {42} \right) - {{\left( {18} \right)}^2}}}\\ = 0.000\end{array}\).

Thus, the regression equation becomes

\(\hat y = 2.00 - 0.00x\).

c.

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

Thus, the presence of an extreme data pair (10,10) can greatly influence the regression equation.