Chapter 2.1, Problem 103E

To determine

To determine:

a) We can see from the table that with the increase in value of $x$ , there is decrease in value of $y$ , hence we can say that coefficient of correlation is negative

b) Equation of least squares is $y=-3.8856x+9.3251$ and correlation coefficient $r=-0.9996$

Plot is

c) Predicted value of y at $x=2.4$ is $0.00034$

Given information: Five points

$x$	1	3	5	7	10
$y$	−5.8	−2.4	−10.7	−17.8	−29.3

Formula used:Slope, $m=\frac{\sum _{i=1}^n(x_i-\overline{X})(y_i-\overline{Y})}{\sum _{i=1}^n{(x_i-\overline{X})}^2}$

Y axis intercept, $b=\overline{Y}-m\overline{X}$

Correlation Coefficient,$r=\frac{\sum _{i=1}^n(x_i-\overline{X})(y_i-\overline{Y})}{\sqrt{\sum _{i=1}^n{(x_i-\overline{X})}^2}\sqrt{\sum _{i=1}^n{(y_i-\overline{Y})}^2}}$

Where $x_i$ and $y_i$ are the $i$ th entry of $x$ and $y$ ; $\overline{X}$ and $\overline{Y}$ are means of $x$ and $y$

Calculation:

Step 1: Calculate the mean of $x$ and $y$

$\begin{array}{â}\overline{X}=\frac{1+3+5+7+10}{5}=5.2\\ \overline{Y}=\frac{5.8+(-2.4)+(-10.7)+(-17.8)+(-29.3)}{5}=-10.88\end{array}$

Step 2: Plot the table as shown

$i$	$x_i$	$y_i$	$x_i-\overline{X}$	$y_i-\overline{Y}$	${(x_i-\overline{X})}^2$	$(x_i-\overline{X})(y_i-\overline{Y})$	${(y_i-\overline{Y})}^2$
1	1	5.8	-4.2	16.68	17.64	-70.056	278.2224
2	3	-2.4	-2.2	8.48	4.84	-18.658	71.9104
3	5	-10.7	-0.2	0.18	0.04	-0.036	0.0324
4	7	-17.8	1.8	-6.92	3.24	-12.456	47.8864
5	10	-29.3	4.8	-18.42	23.04	-88.416	339.2964
					${\displaystyle \sum _{i=1}^n{(x_i-\overline{X})}^2}=48.8$	${\displaystyle \sum _{i=1}^n(x_i-\overline{X})(y_i-\overline{Y})}=-189.62$	${\displaystyle \sum _{i=1}^n{(y_i-\overline{Y})}^2}=737.3480$

Step 3: Calculate the slope $m=\frac{\sum _{i=1}^n(x_i-\overline{X})(y_i-\overline{Y})}{\sum _{i=1}^n{(x_i-\overline{X})}^2}=\frac{-189.62}{48.8}\approx -3.8856$

Step 4: Calculate $y$ intercept $b=\overline{Y}-m\overline{X}=-10.88-(-3.8856\times 5.2)\approx 9.3251$

The slope of the line is $-3.8856$ and y intercept is $9.3251$

Using slope intercept form, $y=mx+b$ ,equation is $y=-3.8856x+9.3251$

Step 5: Draw the scatter plot

Step 6: Calculate the Correlation coefficient

$r=\frac{\sum _{i=1}^n(x_i-\overline{X})(y_i-\overline{Y})}{\sqrt{\sum _{i=1}^n{(x_i-\overline{X})}^2}\sqrt{\sum _{i=1}^n{(y_i-\overline{Y})}^2}}=\frac{-189.62}{\sqrt{48.8}\times \sqrt{737.3480}}\approx -0.9996$

Step 7: Prediction for $x=2.4$ ; plug $x=2.4$ in $y=-3.8856x+9.3251$

$y=-3.8856\left(2.4\right)+9.3251=0.00034$