Multiple Regression & Prediction

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Data

Student Midterm
Christopher J. Zappe, Ph.D.: Out of 100 possible points.		 Final
Christopher J. Zappe, Ph.D.: Out of 100 possible points.
1 72 70
2 95 100 a)
3 83 88
4 85 75
5 65 73
6 82 86
7 84 85
8 74 69
9 74 76
10 82 73
11 81 71
12 75 80
13 78 72
14 78 83
15 72 68
16 73 83
17 75 76
18 79 69
19 72 76
20 75 79 b)
21 77 74 SUMMARY OUTPUT
22 71 80
23 78 70 Regression Statistics
24 83 93 Multiple R 0.7612808123
25 84 78 R Square 0.5795484752
26 71 81 Adjusted R Square 0.5750755867
27 81 73 Standard Error 6.2204935183
28 82 88 Observations 96
29 79 83
30 71 69 ANOVA
31 73 77 df SS MS F Significance F
32 89 87 Regression 1 5013.6195264891 5013.6195264891 129.569173759 2.20852061955901E-19
33 74 74 Residual 94 3637.2867235109 38.6945396118
34 75 78 Total 95 8650.90625
35 93 96
36 74 84 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
37 88 90 Intercept 0.9079168276 6.9800359812 0.1300733735 0.8967863222 -12.9511082382 14.7669418934 -12.9511082382 14.7669418934
38 83 89 Midterm 0.996960229 0.0875844463 11.3828455915 2.20852061955899E-19 0.8230592576 1.1708612003 0.8230592576 1.1708612003
39 90 100
40 82 82
41 79 85 Model equation: Predicted Final = 0.908 + 0.997Midterm
42 62 60
43 73 70
44 88 88
45 76 80
46 76 86
47 76 68 c)
48 80 84
49 84 92
50 84 74
51 91 98
52 70 64
53 76 78
54 74 79
55 68 76
56 80 88
57 87 94 d)
58 92 86
59 84 87
60 79 75
61 80 75
62 91 84
63 74 77
64 69 62
65 88 92
66 84 79
67 83 82 e)
68 87 81
69 82 89
70 72 63
71 97 87
72 88 91
73 70 71
74 83 78
75 92 97
76 94 92
77 63 56
78 83 91
79 81 78
80 84 94
81 78 71
82 84 93
83 91 88
84 78 88
85 71 76
86 77 79
87 78 78
88 70 74
89 86 89
90 79 85
91 70 60
92 75 78
93 80 84
94 75 66
95 77 80 f) Cumulative probabilities
96 74 65 Observation Predicted Final Residuals Ordered residuals Rank (j) (j - 0.5)/n z Score
1 72.6890533137 -2.6890533137 -10.6951328558 1 0.0052083333 -2.5616819349
2 95.6191385801 4.3808614199 -10.6677749165 2 0.015625 -2.1538746941
3 83.6556158324 4.3443841676 -10.6616953745 3 0.0260416667 -1.9424443284
4 85.6495362904 -10.6495362904 -10.6525760614 4 0.0364583333 -1.7933519429
5 65.7103317109 7.2896682891 -10.6495362904 5 0.046875 -1.6759397228
6 82.6586556034 3.3413443966 -10.6130590381 6 0.0572916667 -1.5779228566
7 84.6525760614 0.3474239386 -9.6890533137 7 0.0677083333 -1.493078397
8 74.6829737716 -5.6829737716 -9.6829737716 8 0.078125 -1.417797138
9 74.6829737716 1.3170262284 -9.6799340006 9 0.0885416667 -1.3497900626
10 82.6586556034 -9.6586556034 -9.6586556034 10 0.0989583333 -1.2875097695
11 81.6616953745 -10.6616953745 -8.6768942296 11 0.109375 -1.2298587592
12 75.6799340006 4.3200659994 -8.6708146875 12 0.1197916667 -1.1760288858
13 78.6708146875 -6.6708146875 -8.6616953745 13 0.1302083333 -1.1254068572
14 78.6708146875 4.3291853125 -7.7164112529 14 0.140625 -1.077515567
15 72.6890533137 -4.6890533137 -7.6981726268 15 0.1510416667 -1.0319760584
16 73.6860135427 9.3139864573 -7.6708146875 16 0.1614583333 -0.9884819571
17 75.6799340006 0.3200659994 -7.6312976642 17 0.171875 -0.9467817563
18 79.6677749165 -10.6677749165 -6.6951328558 18 0.1822916667 -0.9066662174
19 72.6890533137 3.3109466863 -6.6708146875 19 0.1927083333 -0.8679592035
20 75.6799340006 3.3200659994 -6.6525760614 20 0.203125 -0.8305108782
21 77.6738544586 -3.6738544586 -6.6434567483 21 0.2135416667 -0.7941925699
22 71.6920930847 8.3079069153 -6.6282578932 22 0.2239583333 -0.7588928329
23 78.6708146875 -8.6708146875 -5.6829737716 23 0.234375 -0.7245143835
24 83.6556158324 9.3443841676 -5.6647351455 24 0.2447916667 -0.690971688
25 84.6525760614 -6.6525760614 -5.6556158324 25 0.2552083333 -0.6581890398
26 71.6920930847 9.3079069153 -5.6525760614 26 0.265625 -0.6260990123
27 81.6616953745 -8.6616953745 -4.6890533137 27 0.2760416667 -0.594641201
28 82.6586556034 5.3413443966 -4.6677749165 28 0.2864583333 -0.5637611914
29 79.6677749165 3.3322250835 -3.6860135427 29 0.296875 -0.5334097062
30 71.6920930847 -2.6920930847 -3.6738544586 30 0.3072916667 -0.5035418943
31 73.6860135427 3.3139864573 -3.6616953745 31 0.3177083333 -0.4741167333
32 89.6373772063 -2.6373772063 -3.6312976642 32 0.328125 -0.445096525
33 74.6829737716 -0.6829737716 -2.719451024 33 0.3385416667 -0.4164464656
34 75.6799340006 2.3200659994 -2.6920930847 34 0.3489583333 -0.3881342775
35 93.6252181222 2.3747818778 -2.6890533137 35 0.359375 -0.3601298918
36 74.6829737716 9.3170262284 -2.6373772063 36 0.3697916667 -0.3324051725
37 88.6404169773 1.3595830227 -2.6221783511 37 0.3802083333 -0.304933676
38 83.6556158324 5.3443841676 -1.6556158324 38 0.390625 -0.2776904398
39 90.6343374352 9.3656625648 -0.6829737716 39 0.4010416667 -0.2506517946
40 82.6586556034 -0.6586556034 -0.6708146875 40 0.4114583333 -0.223795198
41 79.6677749165 5.3322250835 -0.6586556034 41 0.421875 -0.1970990843
42 62.719451024 -2.719451024 -0.6404169773 42 0.4322916667 -0.1705427294
43 73.6860135427 -3.6860135427 0.3048671442 43 0.4427083333 -0.1441061269
44 88.6404169773 -0.6404169773 0.3200659994 44 0.453125 -0.1177698746
45 76.6768942296 3.3231057704 0.3474239386 45 0.4635416667 -0.0915150688
46 76.6768942296 9.3231057704 1.3170262284 46 0.4739583333 -0.0653232053
47 76.6768942296 -8.6768942296 1.3231057704 47 0.484375 -0.0391760855
48 80.6647351455 3.3352648545 1.3261455414 48 0.4947916667 -0.0130557265
49 84.6525760614 7.3474239386 1.3595830227 49 0.5052083333 0.0130557265
50 84.6525760614 -10.6525760614 2.3170262284 50 0.515625 0.0391760855
51 91.6312976642 6.3687023358 2.3200659994 51 0.5260416667 0.0653232053
52 70.6951328558 -6.6951328558 2.3200659994 52 0.5364583333 0.0915150688
53 76.6768942296 1.3231057704 2.3261455414 53 0.546875 0.1177698746
54 74.6829737716 4.3170262284 2.3474239386 54 0.5572916667 0.1441061269
55 68.7012123978 7.2987876022 2.3535034807 55 0.5677083333 0.1705427294
56 80.6647351455 7.3352648545 2.3595830227 56 0.578125 0.1970990843
57 87.6434567483 6.3565432517 2.3747818778 57 0.5885416667 0.223795198
58 92.6282578932 -6.6282578932 3.3048671442 58 0.5989583333 0.2506517946
59 84.6525760614 2.3474239386 3.3109466863 59 0.609375 0.2776904398
60 79.6677749165 -4.6677749165 3.3139864573 60 0.6197916667 0.304933676
61 80.6647351455 -5.6647351455 3.3200659994 61 0.6302083333 0.3324051725
62 91.6312976642 -7.6312976642 3.3231057704 62 0.640625 0.3601298918
63 74.6829737716 2.3170262284 3.3322250835 63 0.6510416667 0.3881342775
64 69.6981726268 -7.6981726268 3.3352648545 64 0.6614583333 0.4164464656
65 88.6404169773 3.3595830227 3.3352648545 65 0.671875 0.445096525
66 84.6525760614 -5.6525760614 3.3413443966 66 0.6822916667 0.4741167333
67 83.6556158324 -1.6556158324 3.3595830227 67 0.6927083333 0.5035418943
68 87.6434567483 -6.6434567483 4.3079069153 68 0.703125 0.5334097062
69 82.6586556034 6.3413443966 4.3170262284 69 0.7135416667 0.5637611914
70 72.6890533137 -9.6890533137 4.3200659994 70 0.7239583333 0.594641201
71 97.6130590381 -10.6130590381 4.3291853125 71 0.734375 0.6260990123
72 88.6404169773 2.3595830227 4.3443841676 72 0.7447916667 0.6581890398
73 70.6951328558 0.3048671442 4.3717421068 73 0.7552083333 0.690971688
74 83.6556158324 -5.6556158324 4.3808614199 74 0.765625 0.7245143835
75 92.6282578932 4.3717421068 5.3322250835 75 0.7760416667 0.7588928329
76 94.6221783511 -2.6221783511 5.3322250835 76 0.7864583333 0.7941925699
77 63.7164112529 -7.7164112529 5.3413443966 77 0.796875 0.8305108782
78 83.6556158324 7.3443841676 5.3443841676 78 0.8072916667 0.8679592035
79 81.6616953745 -3.6616953745 6.3413443966 79 0.8177083333 0.9066662174
80 84.6525760614 9.3474239386 6.3565432517 80 0.828125 0.9467817563
81 78.6708146875 -7.6708146875 6.3687023358 81 0.8385416667 0.9884819571
82 84.6525760614 8.3474239386 7.2896682891 82 0.8489583333 1.0319760584
83 91.6312976642 -3.6312976642 7.2987876022 83 0.859375 1.077515567
84 78.6708146875 9.3291853125 7.3352648545 84 0.8697916667 1.1254068572
85 71.6920930847 4.3079069153 7.3443841676 85 0.8802083333 1.1760288858
86 77.6738544586 1.3261455414 7.3474239386 86 0.890625 1.2298587592
87 78.6708146875 -0.6708146875 8.3079069153 87 0.9010416667 1.2875097695
88 70.6951328558 3.3048671442 8.3474239386 88 0.9114583333 1.3497900626
89 86.6464965193 2.3535034807 9.3079069153 89 0.921875 1.417797138
90 79.6677749165 5.3322250835 9.3139864573 90 0.9322916667 1.493078397
91 70.6951328558 -10.6951328558 9.3170262284 91 0.9427083333 1.5779228566
92 75.6799340006 2.3200659994 9.3231057704 92 0.953125 1.6759397228
93 80.6647351455 3.3352648545 9.3291853125 93 0.9635416667 1.7933519429
94 75.6799340006 -9.6799340006 9.3443841676 94 0.9739583333 1.9424443284
95 77.6738544586 2.3261455414 9.3474239386 95 0.984375 2.1538746941
96 74.6829737716 -9.6829737716 9.3656625648 96 0.9947916667 2.5616819349

Scatterplot

Final 72 95 83 85 65 82 84 74 74 82 81 75 78 78 72 73 75 79 72 75 77 71 78 83 84 71 81 82 79 71 73 89 74 75 93 74 88 83 90 82 79 62 73 88 76 76 76 80 84 84 91 70 76 74 68 80 87 92 84 79 80 91 74 69 88 84 83 87 82 72 97 88 70 83 92 94 63 83 81 84 78 84 91 78 71 77 78 70 86 79 70 75 80 75 77 74 70 100 88 75 73 86 85 69 76 73 71 80 72 83 68 83 76 69 76 79 74 80 70 93 78 81 73 88 83 69 77 87 74 78 96 84 90 89 100 82 85 60 70 88 80 86 68 84 92 74 98 64 78 79 76 88 94 86 87 75 75 84 77 62 92 79 82 81 89 63 87 91 71 78 97 92 56 91 78 94 71 93 88 88 76 79 78 74 89 85 60 78 84 66 80 65

Midterm Score

Final Score

Residual Plot

72 95 83 85 65 82 84 74 74 82 81 75 78 78 72 73 75 79 72 75 77 71 78 83 84 71 81 82 79 71 73 89 74 75 93 74 88 83 90 82 79 62 73 88 76 76 76 80 84 84 91 70 76 74 68 80 87 92 84 79 80 91 74 69 88 84 83 87 82 72 97 88 70 83 92 94 63 83 81 84 78 84 91 78 71 77 78 70 86 79 70 75 80 75 77 74 -2.6890533137014785 4.3808614198951972 4.3443841675838826 -10.649536290364225 7.2896682891169178 3.3413443965579432 0.34742393860983611 -5.6829737716495998 1.3170262283504002 -9.6586556034420568 -10.661695374467996 4.3200659993763395 -6.6708146875458283 4.3291853124541717 -4.6890533137014785 9.3139864573244608 0.32006599937633951 -10.667774916519889 3.3109466862985215 3.3200659993763395 -3.6738544585717676 8.307906915272568 -8.6708146875458283 9.3443841675838826 -6.6525760613901639 9.307906915272568 -8.6616953744679961 5.3413443965579432 3.332225083480111 -2.692093084727432 3.3139864573244608 -2.637377206260453 -0.68297377164959983 2.3200659993763395 2.3747818778433185 9.3170262283504002 1.3595830227136076 5.3443841675838826 9.3656625647654863 -0.65865560344205676 5.332225083480111 -2.7194510239609144 -3.6860135426755392 -0.64041697728639235 3.323105770402293 9.323105770402293 -8.676894229597707 3.3352648545060646 7.3474239386098361 -10.652576061390164 6.3687023357914256 -6.6951328557533714 1.323105770402293 4.3170262283504002 7.29878760219475 7.3352648545060646 6.3565432516876541 -6.6282578931826208 2.3474239386098361 -4.667774916519889 -5.6647351454939354 -7.6312976642085744 2.3170262283504002 -7.6981726267793107 3.3595830227136076 -5.6525760613901639 -1.6556158324161174 -6.6434567483123459 6.3413443965579432 -9.6890533137014785 -10.61305903805291 2.3595830227136076 0.30486714424662864 -5.6556158324161174 4.3717421068173792 -2.6221783511307422 -7.7164112529349751 7.3443841675838826 -3.6616953744679961 9.3474239386098361 -7.6708146875458283 8.3474239386098361 -3.6312976642085744 9.3291853124541717 4.307906915272568 1.3261455414282324 -0.67081468754582829 3.3048671442466286 2.3535034806617148 5.332225083480111 -10.695132855753371 2.3200659993763395 3.3352648545060646 -9.6799340006236605 2.3261455414282324 -9.6829737716495998

Midterm

Residuals

Normal Probability Plot

z Score -10.695132855753371 -10.667774916519889 -10.661695374467996 -10.652576061390164 -10.649536290364225 -10.61305903805291 -9.6890533137014785 -9.6829737716495998 -9.6799340006236605 -9.6586556034420568 -8.676894229597707 -8.6708146875458283 -8.6616953744679961 -7.7164112529349751 -7.6981726267793107 -7.6708146875458283 -7.6312976642085744 -6.6951328557533714 -6.6708146875458283 -6.6525760613901639 -6.6434567483123459 -6.6282578931826208 -5.6829737716495998 -5.6647351454939354 -5.6556158324161174 -5.6525760613901639 -4.6890533137014785 -4.667774916519889 -3.6860135426755392 -3.6738544585717676 -3.6616953744679961 -3.63 12976642085744 -2.7194510239609144 -2.692093084727432 -2.6890533137014785 -2.637377206260453 -2.6221783511307422 -1.6556158324161174 -0.68297377164959983 -0.67081468754582829 -0.65865560344205676 -0.64041697728639235 0.30486714424662864 0.32006599937633951 0.34742393860983611 1.3170262283504002 1.323105770402293 1.3261455414282324 1.3595830227136076 2.3170262283504002 2.3200659993763395 2.3200659993763395 2.3261455414282324 2.3474239386098361 2.3535034806617148 2.3595830227136076 2.3747818778433185 3.3048671442466286 3.3109466862985215 3.3139864573244608 3.3200659993763395 3.323105770402293 3.332225083480111 3.3352648545060646 3.3352648545060646 3.3413443965579432 3.3595830227136076 4.307906915272568 4.3170262283504002 4.3200659993763395 4.3291853124541717 4.3443841675838826 4.3717421068173792 4.3808614198951972 5.332225083480111 5.332225083480111 5.3413443965579432 5.3443841675838826 6.3413443965579432 6.3565432516876541 6.3687023357914256 7.2896682891169178 7.29878760219475 7.3352648545060646 7.3443841675838826 7.3474239386098361 8.307906915272568 8.3474239386098361 9.307906915272568 9.3139864573244608 9.3170262283504002 9.323105770402293 9.3291853124541717 9.3443841675838826 9.3474239386098361 9.3656625647654863 -2.5616819349340227 -2.1538746940614555 -1.942444328423899 -1.7933519429410234 -1.6759397227734436 -1.5779228566327843 -1.493078397012896 -1.4177971379962682 -1.3497900626065398 -1.2875097694894846 -1.229858759216589 -1.1760288858444803 -1.1254068571636224 -1.0775155670402805 -1.0319760584080309 -0.98848195705438602 -0.94678175630104722 -0.90666621739272779 -0.86795920347694333 -0.83051087820540004 -0.79419256994424303 -0.75889283286177922 -0.72451438349236563 -0.69097168796251685 -0.65818903978420318 -0.62609901234642129 -0.59464120100069728 -0.56376119138572522 -0.5334097062412807 -0.50354189431050245 -0.47411673328667442 -0.44509652498551644 -0.41644646561004894 -0.38813427753740326 -0.36012989178956933 -0.33240517246377799 -0.30493367604568428 -0.27769043982157676 -0.25065179462700349 -0.22379519798005867 -0.19709908429431236 -0.17054272938394979 -0.1441061268887168 -0.11776987457909528 -9.151506876835544E-2 -6.532320526972743E-2 -3.9176085503097632E-2 -1.3055726483322334E-2 1.3055726483322472E-2 3.9176085503097632E-2 6.5323205269727291E-2 9.1515068768355579E-2 0.11776987457909528 0.14410612688871671 0.17054272938394993 0.19709908429431236 0.22379519798005856 0.25065179462700365 0.27769043982157676 0.30493367604568405 0.3324051724637781 0.36012989178956933 0.38813427753740304 0.41644646561004917 0.44509652498551644 0.47411673328667431 0.50354189431050256 0.5334097062412807 0.563761191385725 0.59464120100069717 0.62609901234642129 0.65818903978420296 0.69097168796251685 0.72451438349236563 0.75889283286177922 0.79419256994424303 0.83051087820540004 0.86795920347694333 0.90666621739272779 0.94678175630104722 0.98848195705438602 1.0319760584080309 1.0775155670402805 1.1254068571636224 1.1760288858444805 1.229858759216589 1.2875097694894846 1.3497900626065391 1.4177971379962682 1.4930783970128962 1.577922856632785 1.6759397227734436 1.793351942941023 1.9424443284238995 2.1538746940614555 2.5616819349340214

Residuals

Z Score

There is a fairly strong positive relationship, and it is clearly linear.

When the midterm increases by a point, we expect the final exam to increase by nearly a point (0.997).

Note: Model assumptions (1) The model is linear. (2) The error terms have constant variances. (3) The error terms are independent. (4) The error terms are normally distributed.

Note: To generate this residual plot, check the box of residual plot when performing Regression Analysis in the data analysis toolpak.

Note: This Table is generated using Excel Data Analysis ToolPak.

There is no indication of a problem with the assuptions of equal variance and independence (no obvious pattern observed on the residual plot). There is no apparent violation of model assumptions.

Note: To find out these residuals, check the box of residual plot when performing Regression Analysis in the data analysis toolpak.

Residuals are normally distributed.

Note: A straight line, particularly at the middle, indicates that the residuals are normally distributed.