Chapter 12.1, Problem 31E

(a)

To determine

To build: an effective scatterplot to support and clarify this question.

Answer to Problem 31E

Strong positive linear relation

Explanation of Solution

Given:

  

Graph:

"Words" appear on the horizontal and on the vertical side "colours" 

  

Direction: positive, as the scattering ways up.

Form: regular, since it seems that the points lie on a line loosely.

Strength: Mild, as the points are not very distinct, but not close.

The scattering implies therefore a mild positive linear relation between Words and Colours.

(b)

To determine

To find: the equation and describe some variables on the least-square regression line.

Answer to Problem 31E

  $\widehat{y}=4.887+1.132x$

Explanation of Solution

Given:

  

Formula used:

  $\begin{array}{c}a=\frac{{\displaystyle \sum Y\cdot {\displaystyle \sum X^2}-{\displaystyle \sum X\cdot {\displaystyle \sum XY}}}}{n\cdot {\displaystyle \sum X^2}-{\left({\displaystyle \sum X}\right)}^2}\\ b=\frac{n\cdot {\displaystyle \sum XY}-{\displaystyle \sum X\cdot {\displaystyle \sum Y}}}{n\cdot {\displaystyle \sum X^2}-{\left({\displaystyle \sum X}\right)}^2}\end{array}$

Calculation:

Finding the X.Y and ${\text{X}}^2$ .

X	Y	X.Y	X.X
13	20	260	169
10	21	210	100
15	22	330	225
12	25	300	144
13	17	221	169
11	13	143	121
14	32	448	196
16	21	336	256
10	16	160	100
9	13	117	81
11	11	121	121
17	26	442	289
15	20	300	225
15	15	225	225
12	18	216	144
10	18	180	100
${\displaystyle \sum X}$ = 203	${\displaystyle \sum Y}$ =308	${\displaystyle \sum X.Y}$ =4009	${\displaystyle \sum X.X}$ =2665

Finding the value of a and b

  $\begin{array}{c}a=\frac{{\displaystyle \sum Y\cdot {\displaystyle \sum X^2}-{\displaystyle \sum X\cdot {\displaystyle \sum XY}}}}{n\cdot {\displaystyle \sum X^2}-{\left({\displaystyle \sum X}\right)}^2}\\ =\frac{308\cdot 2665-203\cdot 4009}{16\cdot 2665-{203}^2}\\ =4.887\\ b=\frac{n\cdot {\displaystyle \sum XY}-{\displaystyle \sum X\cdot {\displaystyle \sum Y}}}{n\cdot {\displaystyle \sum X^2}-{\left({\displaystyle \sum X}\right)}^2}\\ =\frac{16\cdot 4009-203\cdot 308}{16\cdot 2665-{\left(203\right)}^2}\\ =1.132\end{array}$

Substituting the value, a and b in the equation

  $\begin{array}{c}y=a+b\cdot x\\ y=4.887+1.132\cdot x\end{array}$

The least-squares regression equation is then:

  $\widehat{y}=4.887+1.132x$

With $\widehat{y}$ the expected time for the colours and x the time for the words

  

(c)

To determine

To find: and interpret the student who finished the word assignment in 9 seconds.

Answer to Problem 31E

RESIDUAL=-2.075

Explanation of Solution

Given:

  $\begin{array}{c}x=9\\ y=13\left(\text{since the subject10}\right)\end{array}$

From the part (b)

  $\widehat{y}=4.887+1.132x$

  $\widehat{y}=4.887+1.132x$

Putting the 9 of the x

  $\widehat{y}=4.887+1.132\left(9\right)=15.075$

The residual value reduced by the expected value is the actual value: $\text{RESIDUAL = y - }\widehat{y}=13-15.075=-2.075$

For the student completing the job in 9 seconds, the expected number of colours was overestimated by 2,075 over the real number of colours.

(d)

To determine

To Explain: The P-value.

Explanation of Solution

Given:

  $\begin{array}{l}{H}_0:\beta =0\\ H_a:\beta >0\\ P=0.0215=2.15\%\end{array}$

The P-value is the likelihood of the test statistical value or If the null hypothesis is valid, a value more extreme. If the population regression is zero, the likelihood is small to get an extreme (slope-wise) sample close to the sample providedSample size is 2.15%.