Chapter 4, Problem 10PT

(a)

To determine

Towrite:The equation of best fit line for the data given in table.

Answer to Problem 10PT

The equation of the best fit line is $Y=1.88X+23.624$ .

Explanation of Solution

Given:

The table is:

  

Calculation:

The equation for the regression line can be calculated by performing the following calculations.

Sum of x = 51

Sum of y = 214

Mean x = 10.2

Mean y = 42.8

Sum of squares (SSx) = 230.8

Sum of products (SP) = 435.2

Regression Equation = $\text{Y }=bX+a$

Here,

  $\begin{array}{l}\begin{array}{l}b=\frac{SP}{SSx}=\frac{435.2}{230.8}=\text{ 1}\text{.88}\\ a=\left(Meany\right)-b\left(Meanx\right)=\text{ 42}\text{.8 }-\left(1.88\times 10.2\right)=\text{ 23}\text{.624}\end{array}\hfill \\ \hfill \end{array}$

Substitute the values of a andb .

The equation of the regression line is $Y=1.88X+23.624$

The plot for the regression line is:

  

(b)

To determine

Tofind:The co-relation coefficient and its meaning.

Answer to Problem 10PT

The co-relation coefficient is 0.9844 and its meaning is explained below.

Explanation of Solution

Given:

The table is:

  

Calculation:

The equation for the regression line can be calculated by performing the following calculations.

Sum of x = 51

Sum of y = 214

Mean x = 10.2

Mean y = 42.8

Sum of squares (SSx) = 230.8

Sum of products (SP) = 435.2

Regression Equation = $\text{Y }=bX+a$

Here,

  $\begin{array}{l}\begin{array}{l}b=\frac{SP}{SSx}=\frac{435.2}{230.8}=\text{ 1}\text{.88}\\ a=\left(Meany\right)-b\left(Meanx\right)=\text{ 42}\text{.8 }-\left(1.88\times 10.2\right)=\text{ 23}\text{.624}\end{array}\hfill \\ \hfill \end{array}$

Substitute the values of a andb .

The equation of the regression line is $Y=1.88X+23.624$

The plot for the regression line is:

  

Calculate the co-relation coefficient.

  $\begin{array}{l}XValues\hfill \\ \sum =51\hfill \\ Mean=10.2\hfill \\ \sum {\left(X-M_x\right)}^2=S{S}_x=230.8\hfill \\ \hfill \end{array}$

  $\begin{array}{l}YValues\hfill \\ \sum =214\hfill \\ Mean=42.8\hfill \\ \sum \left(Y-M_y\right)2=S{S}_y=846.8\hfill \end{array}$

  $\begin{array}{l}XandYCombined\hfill \\ N=5\hfill \\ \sum \left(X-M_x\right)\left(Y-M_y\right)=435.2\hfill \end{array}$

  $\begin{array}{l}RCalculation\hfill \\ r=\frac{{\displaystyle \sum \left(\left(X-M_y\right)\left(Y-M_x\right)\right)}}{\sqrt{\left(\left(S{S}_x\right)\left(S{S}_y\right)\right)}}\hfill \\ \hfill \\ r=\frac{435.2}{\sqrt{\left(\left(230.8\right)\left(846.8\right)\right)}}=0.9844\hfill \end{array}$

Therefore the co-relation coefficient is very near to 1. This explains that the fit obtained is almost accurate.

From the co-relation coefficient it can be understood that there is direct co-relation between the years of experience and the salary. The more the number of years of experience, the more the salary is likely to be.