Data Session 7: Solutions

A B C D
Homework

Solutions for Session 7, Part D

See solutions for Problems: D1 | D2 | D3 | D4 | D5 | D6| D7 | D8

Problem D1

Overall, there is an upward trend; that is, the points generally go up and to the right. This corresponds to the positive association between height and arm span.

<< back to Problem D1

Problem D2

a.	The line does a reasonably good job. Some points are above the line, some are below it, and some are on the line, but all are generally pretty close.
b.	It looks like it may be possible for another line to be, overall, "closer" to these points.

<< back to Problem D2

Problem D3

Answers will vary. The lines Height = Arm Span and Height = Arm Span - 1 each seem to do a good job of dividing the points fairly evenly above and below the line, and matching the overall trend of data. It is difficult to distinguish between them without a more mathematical test. Each is clearly better than Height = Arm Span + 1, which lies above a majority of the points.

<< back to Problem D3

Problem D4

Here is the completed table:

Person #

Arm Span (X)

Height (Y)

YL = X

Error = Y - YL

Distance = |Y - YL|

7	162	170	162	8	8
8	165	166	165	1	1
9	170	170	170	0	0
10	170	167	170	-3	3
11	173	185	173	12	12
12	173	176	173	3	3
13	177	173	177	-4	4
14	177	176	177	-1	1
15	178	178	178	0	0
16	184	180	184	-4	4
17	188	188	188	0	0
18	188	187	188	-1	1
19	188	182	188	-6	6
20	188	181	188	-7	7
21	188	192	188	4	4
22	194	193	194	-1	1
23	196	184	196	-12	12
24	200	186	200	-14	14

<< back to Problem D4

Problem D5

Here is the completed table:

Person #

Arm Span (X)

Height (Y)

YL = X - 1

Error = Y - YL

Distance = |Y - YL|

7	162	170	161	9	9
8	165	166	164	2	2
9	170	170	169	1	1
10	170	167	169	-2	2
11	173	185	172	13	13
12	173	176	172	4	4
13	177	173	176	-3	3
14	177	176	176	0	0
15	178	178	177	1	1
16	184	180	183	-3	3
17	188	188	187	1	1
18	188	187	187	0	0
19	188	182	187	-5	5
20	188	181	187	-6	6
21	188	192	187	5	5
22	194	193	193	0	0
23	196	184	195	-11	11
24	200	186	199	-13	13

For the model YL = X - 1, the total vertical distance is 7 + 4 + ... + 13 = 100. Surprisingly, according to this measure of fit, the two lines are equally good. This suggests that another measure of best fit may be useful.

<< back to Problem D5

Problem D6

Here is the completed table:

Person #

Arm Span (X)

Height (Y)

YL = X

Error = Y - YL

(Error)²
=
(Y - YL)²

13	177	173	177	-4	16
14	177	176	177	-1	1
15	178	178	178	0	0
16	184	180	184	-4	16
17	188	188	188	0	0
18	188	187	188	-1	1
19	188	182	188	-6	36
20	188	181	188	-7	49
21	188	192	188	4	16
22	194	193	194	-1	1
23	196	184	196	-12	144
24	200	186	200	-14	196

<< back to Problem D6

Problem D7

Here is the completed table:

Person #

Arm Span (X)

Height (Y)

YL =
X - 1

Error = Y - YL

(Error)²
=
(Y - YL)²

7	162	170	161	9	81
8	165	166	164	2	4
9	170	170	169	1	1
10	170	167	169	-2	4
11	173	185	172	13	169
12	173	176	172	4	16
13	177	173	176	-3	9
14	177	176	176	0	0
15	178	178	177	1	1
16	184	180	183	-3	9
17	188	188	187	1	1
18	188	187	187	0	0
19	188	182	187	-5	25
20	188	181	187	-6	36
21	188	192	187	5	25
22	194	193	193	0	0
23	196	184	195	-11	121
24	200	186	199	-13	169

The sum of squared errors (SSE) is 49 + 16 + ... + 169 = 772. Since this is less than the sum of squared errors for the line Height = Arm Span (which was 784), the line Height = Arm Span - 1 is a slightly better fit.

<< back to Problem D7

Problem D8

a.	The best model is YL = X - .7, because it has the smallest SSE. The worst model is YL = X + 1, because it has the largest SSE.
b.	As all of these lines have the same slope, if we changed the slope, we might find ways to reduce the SSE.
c.	No, we cannot reduce the SSE to zero unless all the data points lie on a straight line, which these 24 points clearly do not do.

<< back to Problem D8

Session 7 | Notes | Solutions | Video