Chapter 4, Problem AP1.12CPT

To determine

To select the best option from the multiple choice question according to the given question.

Answer to Problem AP1.12CPT

Option (b) is correct.

Explanation of Solution

The correlation coefficient is:

  $r=\frac{1}{n-1}{\displaystyle \sum (\frac{x-\overline{x}}{s_x})(\frac{y-\overline{y}}{s_y})}$

The mean is as:

  $\begin{array}{l}\overline{x}=\frac{{\displaystyle \sum _{i=1}^n{x}_i}}{n}\\ =\frac{2+10+6+4+18}{5}\\ =10\\ \overline{y}=\frac{{\displaystyle \sum _{i=1}^n{y}_i}}{n}\\ =\frac{22+4+14+2-4}{5}\\ =7.6\end{array}$

The standard deviation is as:

  $\begin{array}{l}{s}_x=\sqrt{\frac{{(2-10)}^2+\mathrm{...}+{(18-10)}^2}{5-1}}\\ =6.3246\\ s_y=\sqrt{\frac{{(22-7.6)}^2+\mathrm{...}+{(-4-7.6)}^2}{5-1}}\\ =10.3344\end{array}$

Thus we have,

  

Since there are $n=7$ data values, we can then determine the correlation coefficient r :

  $\begin{array}{l}r=\frac{1}{n-1}{\displaystyle \sum (\frac{x-\overline{x}}{s_x})(\frac{y-\overline{y}}{s_y})}\\ =\frac{1}{5-1}\times (-3.9167)\\ =-0.9792\end{array}$

Least square regression line will be:

  $\begin{array}{l}b=r\times \frac{s_y}{s_x}\\ =(-0.9792)\frac{10.3344}{6.3246}\\ =-1.6\\ a=\overline{y}-b\overline{x}\\ =7.6-(-1.6)\times 10\\ =23.6\\ \widehat{y}=a+bx\\ =-1.6+23.6x\end{array}$

Residual will be as:

The residuals are the difference between the observed and the predicted values. Thus, we have,

  

We then note that the largest value of a residual is $3.65$ which belongs to the second point B.

Thus option (b) is correct option.