Chapter 3.1, Problem 20E

(a)

To determine

To make:a scatterplot for the given data

Answer to Problem 20E

Explanation of Solution

Given:

Calculation:

It is assumed that women’s height (X) will help explain men’s height (Y). So, women’s height (x) is the explanatory variable and men’s height (Y) is the response variable. We want to see men’s height (Y) changes when women’s height (X) changes, so we put women’s height X (the explanatory variable) on the horizontal axis and men’s height Y (the response variable) on the vertical axis.

Using EXCEL, the scatterplot for the given data is,

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 form lower left to upper right. The relationship is moderately strong; we cannot expect correlation near $\pm$ 1 because the points deviate greatly from line. Since scatter plot shows positive relationship between the heights of women and height of men with whom they are dating, the correlation doesn’t help to tell whether women tend to date men taller or shorter than themselves. Instead it says that the men that shorter women tend to date are shorter than the men the taller women tend to date.

Conclusion:

Hence, scatterplot is drawn.

(b)

To determine

To find:the correlation and explain whether the data shows that taller women tend to date taller men.

Answer to Problem 20E

Hence,

the correlation of 0.57 confirms in the scatter plot: there is moderately strong, positive relationship between two variables.

The value of correlation is 0.57

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	${(X-\overline{X})}^2$	${(Y-\overline{Y})}^2$	$\left(\frac{X-\overline{X}}{S_x}\right)$	$\left(\frac{Y-\overline{Y}}{S_y}\right)$	$\left(\frac{X-\overline{X}}{S_x}\right)\left(\frac{Y-\overline{Y}}{S_y}\right)$
	66	72	0.0	9.0	0.00	1.19	0.00
	64	68	4.0	1.0	-0.95	-0.40	0.38
	66	70	0.0	1.0	0.00	0.40	0.00
	65	68	1.0	1.0	-0.48	-0.40	0.19
	70	71	16.0	4.0	1.91	0.79	1.51
	65	65	1.0	16.0	-0.48	-1.58	0.75
Total	396	414	22	32	0.00	0.00	2.83

  ${\displaystyle \sum X=396}$

  ${\displaystyle \sum Y=414}$

  ${\displaystyle \sum {(X-\overline{X})}^2}=22$

  ${\displaystyle \sum {(Y-\overline{Y})}^2}=32$

  ${\displaystyle \sum \left(\frac{X-\overline{X}}{S_x}\right)\left(\frac{Y-\overline{Y}}{S_y}\right)=2.83}$

  $\overline{X}=\frac{1}{6}(396)$

=66.0

  $\overline{Y}=\frac{1}{6}(414)$

=69.0

  $S_x=\sqrt{\frac{1}{6-1}(22)}$

=2.1

  $S_x=\sqrt{\frac{1}{6-1}(32)}$

=2.5

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

  $=\frac{2.83}{6-1}$

= 0.57

The correlation of 0.57 confirms in the scatter plot: there is moderately strong, positive relationship between two variables. Removing the outlier (70, 71) would increase the correlation because remaining 5 points are tightly clustered in a linear pattern.

Conclusion:

Hence, the correlation of 0.57 confirms in the scatter plot: there is moderately strong, positive relationship between two variables.

The value of correlation is 0.57