3/5/07 251grass2
(Open this document in 'Page Layout' view!) Graded Assignment 2
Name:
Class days and time:
Student number:
There will be a penalty for papers that are unstapled or do not have the three information items
requested above. Note that from now on neatness means paper neatly trimmed on the left side if it
has been torn, multiple pages stapled and paper written on only one side. The stapling is for your
protection – putting your name on every page helps too, I still have some unclaimed pages from an
old exam (as well as an old exam with no name on it that the perp will not admit responsibility for).
1) Use the second joint probability table in Problem K4. Modify the table as follows: subtract the last digit
of your student number (divided by 100) from all three numbers on the diagonal, add the same number to
any 3 numbers off the diagonal, if the last digit of your student number is zero, use 10. For example, if the
last two digits of your number are 30, the .40 on the diagonal becomes .40 - .10 = .30 and a zero will
become .10. The sum of the numbers in the table will not change.
For this joint probability table (i) check for independence, (ii) Compute E x  and Varx  , (iii) Compute
 
 
Covx, y  or  xy and Corr x, y  or  xy , (iv) Compute Ex  y  and Var x  y  from the results in (ii)
and (iii), (iv) Compute Cov3x  3, y  and Corr 3x  3, y  using the formulas in section K4 of 251v2out
or section C1 of 251var2. Note that y  1y  0 .
2) The following data represent the scores of a group of students on a math placement test and their grades
in a math course. Personalize the data by subtracting the second to last digit of your student number from
the 45 in the x column. (i) Compute the sample mean and variance of x , (ii) Compute Covx, y  or s xy
and Corr x, y  or rxy , (iii) Compute the sample mean and variance of
x  y 
from the results in (i) and
(ii). (iv) Compute Cov6 x  3, y  and Corr 6 x  3, y  using the formulas in section K4 of 251v2out or
section C1 of 251var2. Note that y  1y  0 .
Test Score
Grades
y
x
51
75
52
72
59
82
45
67
61
75
54
79
56
78
67
82
63
87
53
72
60
96
————— 3/9/2007 2:04:45 PM ————————————————————
Welcome to Minitab, press F1 for help.
51
52
59
45
61
54
56
67
63
53
60
75
72
82
67
75
79
78
82
87
72
96
Results for: 1gr2-071.MTW
MTB > WSave "C:\Documents and Settings\RBOVE\My Documents\Minitab\1gr2071.MTW";
SUBC>
Replace.
Saving file as: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\1gr2-071.MTW'
MTB > let c6 = c1
MTB > let c7 = c2
MTB > let c3 = c2*c1
MTB > let c3=c1*c1
MTB > let c4 = c2*c2
MTB > let c5 = c1*c2
MTB
MTB
MTB
MTB
>
>
>
>
let c8 = c6*c6
let c9 = c7*c7
let c10 = c6 * c7
print c1 - c5
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
x
51
52
59
45
61
54
56
67
63
53
60
y
75
72
82
67
75
79
78
82
87
72
96
xsq
2601
2704
3481
2025
3721
2916
3136
4489
3969
2809
3600
ysq
5625
5184
6724
4489
5625
6241
6084
6724
7569
5184
9216
xy
3825
3744
4838
3015
4575
4266
4368
5494
5481
3816
5760
MTB > describe c1 c2
Descriptive Statistics: x, y
Variable
x
y
N
11
11
N*
0
0
Mean
56.45
78.64
SE Mean
1.89
2.42
StDev
6.27
8.03
MTB > corr c1 c2
Correlations: x, y
Pearson correlation of x and y = 0.693
P-Value = 0.018
MTB > Covariance c1 c2.
Covariances: x, y
x
y
x
39.2727
34.8818
y
64.4545
MTB > sum c1
Sum of x
Sum of x = 621
MTB > sum c2
Sum of y
Sum of y = 865
MTB > sum c3
Sum of xsq
Sum of xsq = 35451
MTB > sum c4
Sum of ysq
Sum of ysq = 68665
MTB > sum c5
Sum of xy
Sum of xy = 49182
Minimum
45.00
67.00
Q1
52.00
72.00
Median
56.00
78.00
Q3
61.00
82.00
Maximum
67.00
96.00
Row
1
2
3
4
5
6
7
8
9
10
11
x2
y
x
y2
xy
51 75 2601 5625 3825
52 72 2704 5184 3744
59 82 3481 6724 4838
45 67 2025 4489 3015
61 75 3721 5625 4575
54 79 2916 6241 4266
56 78 3136 6084 4368
67 82 4489 6724 5494
63 87 3969 7569 5481
53 72 2809 5184 3816
60 96 3600 9216 5760
621 865 35451 68665 49182
 x  621,  y  865 ,  x  35451 ,
 x  621  56.4545 and y   y  865  78.6364 .
x
2
To summarize the results of these computations n  11 ,
y
s x2
2
 68665 and
x

2
 nx 2
n 1

 xy  49182 . Thus
n
11
n
11
35451  1156 .4545 2 392 .78373

 39 .2784 . Minitab says 39.2727.
10
10
s x  39 .2784  6.2672
s 2y
y

2
 ny 2
n 1

68665  1178 .6364 2 644 .48254

 64 .4483 . Minitab says 64.4545.
10
10
s x  64 .4483  8.0280
s xy 
 x  x  y  y    xy  nx y
n 1
n 1

49182  1156 .4545 78 .6364  348 .83492

 34 .8835 . Minitab
10
10
says 34.8818
rxy 
s xy
sx s y

34 .8835

39 .2784 64 .4483
34 .8835 2

39 .2784 64 .4483 
0.4807  .6933 . Minitab says .693.
————— 3/21/2007 12:52:31 PM ————————————————————
Welcome to Minitab, press F1 for help.
MTB > WOpen "C:\Documents and Settings\RBOVE\My Documents\Minitab\1gr2071.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\1gr2-071.MTW'
Worksheet was saved on Fri Mar 09 2007
Results for: 1gr2-071a.MTW
MTB > WSave "C:\Documents and Settings\RBOVE\My Documents\Minitab\1gr2071a.MTW";
SUBC>
Replace.
Saving file as: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\1gr2-071a.MTW'
MTB > print c6 - c10
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
x1
51
52
59
36
61
54
56
67
63
53
60
y1
75
72
82
67
75
79
78
82
87
72
96
x1sq
2601
2704
3481
1296
3721
2916
3136
4489
3969
2809
3600
y1sq
5625
5184
6724
4489
5625
6241
6084
6724
7569
5184
9216
x1y1
3825
3744
4838
2412
4575
4266
4368
5494
5481
3816
5760
MTB > sum c6
Sum of x1
Sum of x1 = 612
MTB > sum c7
Sum of y1
Sum of y1 = 865
MTB > ssq c6
Sum of Squares of x1
Sum of squares (uncorrected) of x1 = 34722
MTB > sum c8
Sum of x1sq
Sum of x1sq = 34722
MTB > sum c9
Sum of y1sq
Sum of y1sq = 68665
MTB > sum c10
Sum of x1y1
Sum of x1y1 = 48579
MTB > describe c6 c7
Descriptive Statistics: x1, y1
Variable
x1
y1
N
11
11
N*
0
0
Mean
55.64
78.64
SE Mean
2.47
2.42
StDev
8.20
8.03
MTB > covariance c6 c7
Covariances: x1, y1
x1
y1
x1
67.2545
45.3545
y1
64.4545
MTB > corr c6 c7
Correlations: x1, y1
Pearson correlation of x1 and y1 = 0.689
P-Value = 0.019
Minimum
36.00
67.00
Q1
52.00
72.00
Median
56.00
78.00
Q3
61.00
82.00
Maximum
67.00
96.00
Row
1
2
3
4
5
6
7
8
9
10
11
x2
y
x
y2
xy
51 75 2601 5625 3825
52 72 2704 5184 3744
59 82 3481 6724 4838
36 67 1296 4489 2412
61 75 3721 5625 4575
54 79 2916 6241 4266
56 78 3136 6084 4368
67 82 4489 6724 5494
63 87 3969 7569 5481
53 72 2809 5184 3816
60 96 3600 9216 5760
612 865 34722 68665 48579
 x  612 ,  y  865 ,  x  34722 ,
 x  612  55.6364 and y   y  865  78.6364 .
x
2
To summarize the results of these computations n  11 ,
y
2
 68665 and
x
s x2 
2
 nx 2
n 1

 xy  48579 . Thus
n
11
n
11
34722  1155 .6364 2 672 .50094

 67 .2501 . Minitab says 67.2545.
10
10
s x  67 .2501  8.2006
y
s 2y 
2
 ny 2
n 1

68665  1178 .6364 2 644 .48254

 64 .4483 . Minitab says 64.4545.
10
10
s x  64 .4483  8.0280
s xy 
 x  x  y  y    xy  nx y
n 1
says 45.3545
rxy 
s xy
sx s y
n 1
45 .3492


67 .2501 64 .4483

48579  1155 .6364 78 .6364  453 .49175

 45 .3492 . Minitab
10
10
45.3492 2

67 .2501 64.4483 
0.4745  .6888 . Minitab says .689.
MTB > describe c6 c7
Descriptive Statistics: x1, y1
Variable
x1
y1
N
11
11
N*
0
0
Mean
55.64
78.64
SE Mean
2.47
2.42
StDev
8.20
8.03
Minimum
36.00
67.00
Q1
52.00
72.00
Median
56.00
78.00
Q3
61.00
82.00
MTB > corr c6 c7
Correlations: x1, y1
Pearson correlation of x1 and y1 = 0.689
P-Value = 0.019
MTB > covar c6 c7
Covariances: x1, y1
x1
y1
x1
67.2545
45.3545
y1
64.4545
MTB > Save "C:\Documents and Settings\RBOVE\My Documents\Minitab\1gr2071.MTW";
SUBC>
Replace.
Saving file as: 'C:\Documents and Settings\RBOVE\My
Documents\Minitab\1gr2-071.MTW'
Maximum
67.00
96.00
Existing file replaced.
MTB >
Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
x
y
xsq
ysq
xy
51 75 2601 5625 3825
52 72 2704 5184 3744
59 82 3481 6724 4838
45 67 2025 4489 3015
61 75 3721 5625 4575
54 79 2916 6241 4266
56 78 3136 6084 4368
67 82 4489 6724 5494
63 87 3969 7569 5481
53 72 2809 5184 3816
60 96 3600 9216 5760
621 865 35451 68665 49182
 x  621,  x  35451 ,  y  865 ,  y  68665 ,
 x  621  56.454545 and y   y  865  78.636364 .
Then x 
So n  11,
s x2 
s 2y 
2
n
x
2
y
2
2
11
 nx
 ny 2
n 1
 xy
 49162 .
11

35451  1156 .454545 
 39 .272728 ,
10

68665  1178 .636364 2
 64 .454483 . ( s x  39 .272728  6.2668 and
10
2
n 1
n
and
2
s y  64.454483  8.0284 ).
(ii) Compute Covx, y  or s xy and Corr x, y  or rxy .
s xy  Covx, y  
rxy 
s xy
sx s y
 xy  nxy  49182  1156.454545 78.636364   34.881835
n 1
 Corr x, y  
10
34 .881835
39 .272728
and
 0.4806782  .6933 . The correlation and
64 .454483
covariance are positive, indicating a tendency of y to rise when x rises. rxy2  .4807 is neither large nor
small on a zero to one scale, indicating that the relationship is not terribly strong. Note that 1  rxy  1
always!
Descriptive Statistics: x, y
Variable
x
y
N
11
11
N*
0
0
Mean
56.45
78.64
SE Mean
1.89
2.42
StDev
6.27
8.03
MTB > corr c1 c2
Correlations: x, y
Pearson correlation of x and y = 0.693
P-Value = 0.018
MTB > Covariance c1 c2.
Covariances: x, y
x
y
x
39.2727
34.8818
y
64.4545
Minimum
45.00
67.00
Q1
52.00
72.00
Median
56.00
78.00
Q3
61.00
82.00
Maximum
67.00
96.00