Chapter 3.1, Problem 19E

(a)

To determine

To Explain: all five specimens come from the same species and make a scatterplot

Answer to Problem 19E

  

Explanation of Solution

Given:

  

Calculation:

Using Excel, the scatterplot for the given data is,

  

From the scatter plot, it is observed that the overall pattern moves from lower left to upper right. We call this a positive association between the two variables. The form of the relationship is linear. That is, the overall pattern follows a straight line from lower left to upper right. The relationship is very strong because the points don’t deviate greatly from line. Since scatter plot shows positive relationship between the lengths of a pair of bones from all individuals, we can say that all five specimens come from the same species.

Conclusion:

Hence, the scatter plot shows positive relationship between the lengths of a pair of bones from all the individuals.

(b)

To determine

To find: the correlation and explain how the value of r matches the graph

Answer to Problem 19E

The value of correlation is 0.995

The correlation of 0.995 confirms in the scatter plot: there is very strong, positive relationship between two variables.

Explanation of Solution

Calculation:

We find the correlation coefficient is given as follows:

  $r=\frac{1}{n-1}{\displaystyle \sum _{i=1}^n\left(\frac{X_i-\overline{X}}{S_x}\right)}\left(\frac{Y_i-\overline{Y}}{S_y}\right)$

Where$\overline{X}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{X}_i}$

  $\overline{Y}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{Y}_i}$

  $S_x=\sqrt{\frac{1}{n-1}{\displaystyle \sum _{i=1}^n{(X_i-\overline{X})}^2}}$

  $S_y=\sqrt{\frac{1}{n-1}{\displaystyle \sum _{i=1}^n{(Y_i-\overline{Y})}^2}}$

Consider the following table:

	X	Y	${(X-\overline{X})}^2$	${(Y-\overline{Y})}^2$	$\left(\frac{X-\overline{X}}{S_x}\right)$	$\left(\frac{Y-\overline{Y}}{S_y}\right)$	$\left(\frac{X-\overline{X}}{S_x}\right)\left(\frac{Y-\overline{Y}}{S_y}\right)$
	38	41	408.0	625.0	-1.53	-1.57	2.41
	56	63	4.8	9.0	-0.17	-0.19	0.03
	59	70	0.6	16.0	0.06	0.25	0.02
	64	72	33.6	36.0	0.44	0.38	0.17
	74	84	249.6	324.0	1.20	1.13	1.36
Total	291	330	696.8	1010	0	0	3.98

  ${\displaystyle \sum X=291}$

  ${\displaystyle \sum Y=330}$

  ${\displaystyle \sum {(X-\overline{X})}^2}=696.8$

  ${\displaystyle \sum {(Y-\overline{Y})}^2}=1010$

  ${\displaystyle \sum \left(\frac{X-\overline{X}}{S_x}\right)\left(\frac{Y-\overline{Y}}{S_y}\right)=3.98}$

  $\overline{X}=\frac{1}{5}(291)$

=58.2$\overline{Y}=\frac{1}{5}(330)$

=66.0$S_x=\sqrt{\frac{1}{5-1}(696.8)}$

=13.2$S_y=\sqrt{\frac{1}{5-1}(1010.0)}$

=15.9$r=\frac{1}{n-1}{\displaystyle \sum _{i=1}^n\left(\frac{X_i-\overline{X}}{S_x}\right)}\left(\frac{Y_i-\overline{Y}}{S_y}\right)$

  $=\frac{3.98}{5-1}$

= 0.995

The correlation of 0.995 confirms what we see in the scatter plot: there is very strong, positive relationship between two variables. Removing the outlier (59, 70) would increase the correlation because remaining 4 points are tightly clustered in a linear pattern.

Conclusion:

Hence,

the value of correlation is 0.995

The correlation of 0.995 confirms in the scatter plot: there is very strong, positive relationship between two variables.