Question

The following table is similar to Table 11.2. It is used for making the preliminary computations for finding the least-squares line for the given pairs of x- and y-values.

a. Complete the table.

b. FindSS_xy.

c. Find SS_xx.

d. Find$\widehat{{\text{β}}_{\text{1}}}$.

e. Find$\overline{X}and\overline{Y}$.

f. Find$\widehat{{\text{β}}_{\text{0}}}$.

g. Find the least-squares line.

Short answer

Answer

1. Fig.1 Table
2. -26.29
3. 33.71
4. -0.77
5. 3.43, 4.43
6. 1.78
7. -16.7

Step-by-step solution

Introduction

The Least Squares Regression Line is the line that has the shortest vertical distance between the data points and the regression line. It's dubbed a "least squares" method because the optimum fit line reduces variance.

Complete the table

Find  SSxy

$$\begin{gathered} \begin{array}{l}\begin{array}{l}{\mathbf{\text{SS}}}_{\mathbf{\text{xy}}}\mathbf{\text{=}}\mathbf{\sum }{\mathbf{\text{x}}}_{\mathbf{\text{i}}}{\mathbf{\text{y}}}_{\mathbf{\text{i}}}\mathbf{-}\frac{{\displaystyle \sum {\text{x}}_{\text{i}}\sum {\text{y}}_{\text{i}}}}{\mathbf{\text{n}}}\\ \mathbf{\text{=80}}\mathbf{-}\frac{\mathbf{\text{24x31}}}{\mathbf{\text{7}}}\\ \mathbf{\text{=80}}\mathbf{-}\frac{\mathbf{\text{744}}}{\mathbf{\text{7}}}\\ \mathbf{\text{=}}\frac{\mathbf{\text{560}}\mathbf{-}\mathbf{\text{744}}}{\mathbf{\text{7}}}\end{array} \\ \mathbf{\text{=}}\mathbf{-}\frac{\mathbf{\text{184}}}{\mathbf{\text{7}}}\\ \\ \mathbf{\text{=}}\mathbf{-}\mathbf{\text{26.29}}\end{array} \end{gathered}$$

Find  SSxx

$$\begin{gathered} \begin{array}{l}{\mathbf{\text{SS}}}_{\mathbf{\text{xy}}}\mathbf{\text{=}}\mathbf{\sum }{{\mathbf{\text{x}}}_{\mathbf{\text{i}}}}^{\mathbf{\text{2}}}\mathbf{-}\frac{{\mathbf{\left(}{\displaystyle \sum {\text{x}}_{\text{i}}}\mathbf{\right)}}^{\mathbf{\text{2}}}}{\mathbf{\text{n}}}\\ \mathbf{\text{=116}}\mathbf{-}\frac{{\mathbf{\left(}\mathbf{\text{24}}\mathbf{\right)}}^{\mathbf{\text{2}}}}{\mathbf{\text{7}}}\\ \mathbf{\text{=116}}\mathbf{-}\frac{\mathbf{\text{576}}}{\mathbf{\text{7}}}\\ \mathbf{\text{=}}\frac{\mathbf{\text{812}}\mathbf{-}\mathbf{\text{576}}}{\mathbf{\text{7}}}\end{array} \\ \begin{array}{l}\mathbf{\text{=}}\frac{\mathbf{\text{236}}}{\mathbf{\text{7}}}\\ \mathbf{\text{= 33.71}}\end{array} \end{gathered}$$

Find β1^

$\begin{array}{c}\widehat{{\mathbf{\text{β}}}_{\mathbf{\text{1}}}}\mathbf{\text{=}}\frac{{\mathbf{\text{SS}}}_{\mathbf{\text{xy}}}}{{\mathbf{\text{SS}}}_{\mathbf{\text{xx}}}}\\ \mathbf{\text{=}}\mathbf{-}\frac{\mathbf{\text{26.29}}}{\mathbf{\text{33.71}}}\\ \mathbf{\text{=}}\mathbf{-}\mathbf{\text{0.77}}\end{array}$

Find and X ¯and Y¯

$\begin{array}{l}\overline{\mathbf{\text{x}}}\mathbf{\text{=}}\frac{{\displaystyle \sum {\text{x}}_{\text{i}}}}{\mathbf{\text{n}}}\\ \mathbf{\text{=}}\frac{\mathbf{\text{24}}}{\mathbf{\text{7}}}\\ \mathbf{\text{= 3.43}}\end{array}$

$\begin{array}{c}\overline{\mathbf{\text{y}}}\mathbf{\text{=}}\frac{{\displaystyle \sum {\text{y}}_{\text{i}}}}{\mathbf{\text{n}}}\\ \mathbf{\text{=}}\frac{\mathbf{\text{31}}}{\mathbf{\text{7}}}\\ \mathbf{\text{= 4.43}}\end{array}$

Find β0^

$\begin{array}{c}\widehat{{\mathbf{\text{β}}}_{\mathbf{\text{0}}}}\mathbf{\text{=}}\overline{\mathbf{\text{y}}}\mathbf{-}\widehat{{\mathbf{\text{β}}}_{\mathbf{\text{1}}}}\overline{\mathbf{\text{x}}}\\ \mathbf{\text{= 4.43}}\mathbf{-}\mathbf{(}\mathbf{-}\mathbf{\text{0.77}}\mathbf{\times }\mathbf{\text{3.43}}\mathbf{)}\\ \mathbf{\text{= 4.43 + 2.6411}}\\ \mathbf{\text{= 1.78}}\end{array}$

Find the least-squares line

$\begin{array}{c}\widehat{\mathbf{\text{y}}}\mathbf{\text{=}}\widehat{{\mathbf{\text{β}}}_{\mathbf{\text{0}}}}\mathbf{\text{+}}\widehat{{\mathbf{\text{β}}}_{\mathbf{\text{1}}}}{\mathbf{\text{x}}}_{\mathbf{\text{i}}}\\ \mathbf{\text{=1.78 +}}\mathbf{(}\mathbf{-}\mathbf{\text{0.77}}\mathbf{\times }\mathbf{\text{24}}\mathbf{)}\\ \mathbf{\text{= 1.78 +}}\mathbf{(}\mathbf{-}\mathbf{\text{18.48}}\mathbf{)}\\ \mathbf{\text{=1.78}}\mathbf{-}\mathbf{\text{18.48}}\\ \mathbf{\text{=}}\mathbf{-}\mathbf{\text{16.7}}\end{array}$

Hence, the least-squares line is -16.7.