Chapter 8.1, Problem 28E

a)

To determine

To construct a scatter plot and describe it.

Answer to Problem 28E

There is weak positive correlation between brother’s and sister’s height.

Explanation of Solution

Given:

Brother	Sister
71	69
68	64
66	65
67	63
70	65
71	62
70	65
73	64
72	66
65	59
66	62

The scatter plot is,

  

The correlation is positive as scatter plot shows trend of the data upwards. There seems to be weak correlation because the points are scattered.

b)

To determine

To find the least square regression line.

Answer to Problem 28E

  $y=0.527x+27.635$

Explanation of Solution

Given:

Brother	Sister
71	69
68	64
66	65
67	63
70	65
71	62
70	65
73	64
72	66
65	59
66	62

The least square regression line is,

  $\widehat{y}=0.527x+27.635$

c)

To determine

To interpret slope of the regression equation.

Answer to Problem 28E

On average the height of the sister increases by 0.527 inches per inch of the height of the brother.

Explanation of Solution

Given:

The least square regression line is,

  $\widehat{y}=0.527x+27.635$

The slope is, b= 0.527

The slope is positive. Therefore, on average the height of the sister increases by 0.527 inches per inch of the height of the brother.

d)

To determine

To calculate and interpret the residual for the first pair.

Answer to Problem 28E

we underestimated the height of the sister by 3.9479 inches when making a prediction using the regression line.

Explanation of Solution

Given:

The least square regression line is,

  $\widehat{y}=0.527x+27.635$

Formula:

  $\mathrm{Re}sidual=y-\widehat{y}$

Using regression equation, first we find predicted value by replacing $x=71$

Therefore,

  $\begin{array}{l}\widehat{y}=0.527\times 71+27.635\\ \widehat{y}=65.0521\end{array}$

Observed y = 69

Residual can be found as,

  $\begin{array}{l}\mathrm{Re}sidual=y-\widehat{y}\\ \mathrm{Re}sidual=69-65.0521\\ \mathrm{Re}sidual=3.9479\end{array}$

This can be interpreted as, we underestimated the height of the sister by 3.9479 inches when making a prediction using the regression line.