1
Department of Mathematics and Statistics
Semester 132
STAT310
Midterm Exam
Name:
Thursday, April 17, 2014
ID #:
Question One:
The director of admissions of a small college selected 120 students at random from the new
freshman class in a study to determine whether a student's grade point average (GPA) at the end of the
freshman year (Y) can be predicted from the ACT test score (X). The results of the study follow.
provided: ∑y=, ∑x=, ∑x2 =, ∑y2 =, ∑xy=
1. Fill the missing values “?” below?
The regression equation is:
β1
?
Predictor
Constant
x
Coef
SE Coef
T
________
________
?________ 0.3209
?________ 0.01277
S = ?________
P
0.000
0.003
R-Sq = ?________
β0
Analysis of Variance
Source
Regression
Residual Error
Lack of Fit
Pure Error
Total
DF
SS
?___
____
34
83
____
?_______
45.8176
14.6600
31.1042
________
MS
_______
________
0.4312
0.3747
F
P
9.24 0.003
1.15 0.298
Predicted Values for New Observations
Obs Fit SE Fit
1 3.2012 0.0706
95% CI
95% PI
(3.0614, 3.3410) (1.9594, 4.4431)
Unusual Observations
Obs
2
9
101
102
105
115
SRES1
2.027999
-4.43288
-2.00435
-2.00137
-0.21259
-2.97678
TRES1
2.055526
-4.83496
-2.03071
-2.02758
-0.21172
-3.08213
HI1
0.056665
0.016012
0.008365
0.027336
0.008554
0.024878
COOK1
0.123525
0.159886
0.016945
0.056286
0.000195
0.113038
DFIT1
0.503787
-0.61677
-0.18651
-0.33991
-0.01967
-0.4923
2
Residual Plots for y
Fitted Line Plot
Normal Probability Plot
y = A+B x
50 89 116 1
52
94
3
y
32
48
2
119
74
73
100
22 111
81 37
57
82 30
71
5
49 103 98
35
76
99
28
4
90
113 66
114
58
20
91
96
64 55
70
11
77
36
13
14
72 105
62 67
46
21
43
31
17
93
63
85
19
61
110
107
0
10
0.1
47
-2
0
Residual
-3
2
9
0
30
35
15
10
5
0
-2.25
-1.50
-0.75
0.00
Residual
0.75
1
SRES1
0
-1
-2
50
89
31
91
8
79
96
29 24
118
88
60
75
64
3
117
23
40
55
70 111109
37 95 81 56
17
108
39
25 22
93
112
63
11
34
57
7753
87
82 12
71
85
30
5
10
13
59 49
36
103
26
14
78
729735
98
76 38 99
105
19
41
61
54 16
110
33
1076286
51 120
67
28
90
46
4
21
66
43
113
114
65
58 42
27
20
44
83 92
45
101
6
94
18
74
73
100
47
69
102
52
84
68
104
106
80
15
32
7
119
48
115
-3
-4
9
-5
0.01
0.02
0.03
0.04
3.0
3.2
Fitted Value
3.4
0
-1
-2
2
116
1
2.8
Versus Order
Scatterplot of SRES1 vs HI1
2
2.6
1
Residual
Frequency
1
25
x
115
Histogram
115
20
-1
9
20
15
91
96 31 29 24
8 118
79 60
88 6 84
75
64
3 117
40 109
23
81 37
2211170 55 17
56 68
25 95112
93 34 39108
11 63
12 104
106
94 49 71
82 57
87
30
53
5 77
80
36 85 10
13
59
38
76 98
14
78
26
97
74 99103
35 72105
19
61 41
54 51
110
16
86120
62
107
33
73
18 7 15
32 100
90
28
67
46
11366 4 21
43
114
42
65
48119
58
20
44 92 27
69
45 47
102
10183
52
-2
101
69
102
50 89116 1
1
27
45
1
90
50
65
42
44 92
83
99
Residual
2
Percent
4
Versus Fits
2
99.9
8 79 60
88
75
29 24 118
6 84
109
3 117
95 56
23
68
106
40
108
112
104
39 25
12
80
34 87
59
53
38
15
10 97
26
78
18 7
41
16 120
54 86
33
51
0.05
HI1
2. Does the estimated regression function appear to fit the data well?
0.06
-3
1
10
20
30
40
50
60
70
80
Observation Order
90 100 110 120
3
3. Do outliers and/or leverage points exist? If yes, are the outliers and leverage points valid?
4. What is the point estimate of the change in the mean response when the entrance test score increases by one
point?
5. Test whether or not a linear association exists between student’s ACT score (X) and GPA at the end of the freshman
year (Y). Use a level of significance of 0.01. (State the alternatives, decision rule, and conclusion.)
6. Obtain a 99% confidence interval for β1. Interpret your confidence interval.
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7. Find the 90% joint confidence intervals for β0 and β1 and interpret.
8. Obtain a 95% interval estimate of the mean freshman GPA for students whose ACT test score is 28. Interpret your
confidence interval.
Interval:
Interpretation:
9. Ahmad obtained a score of 28 on the entrance test. Predict his freshman GPA using a 95% prediction interval.
Interpret your prediction interval.
Interval:
Interpretation:
10. Compute R-sq and interpret its value? What can we conclude about the model?
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11. Lack of fit test outputs provided and assume the model assumptions are satisfied:
a. Can you perform a lack of fit test for this data set? Explain
b. Assume yes, perform the test and write your conclusion.
̅, 𝒀
̅ ).
12. Verify that your fitted regression line goes through the point (𝑿
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Part 2:
Consider the data sets in Blackboard. Use Minitab to solve the parts below. Save all Minitab output used in a word file
called “yourLastName-ID”, and email it to me in blackboard or to mailto:[email protected] the end of the
exam session.
Question Two:
(Production time318) In a manufacturing study, the production times for 111 recent production
runs were obtained. The table below lists for each run the production time in hours (7) and the production lot
size (X).
1. Based on the scatter plot of the data: Does a linear relation appear adequate here? Would a transformation on
X or Y be more appropriate here? Why?
2. obtain the estimated linear regression function and
a. Comment on the assumptions.
a. Do outliers and/or leverage points exist? If yes, are the outliers and leverage points valid?
3. Use the transformation X' = √𝐗
a. Repeat part 2.
b. Estimate y when X=4
7
Question Three: (Sales growth317) A marketing researcher studied annual sales of a product that had been
introduced 10 years ago. The data are as follows, where X is the year (coded) and Y is sales in thousands
2
Prepare a scatter plot of the data. Does a linear relation appear adequate here?
3
obtain the estimated linear regression function and
a. Comment on the assumptions.
b. Do outliers and/or leverage points exist? If yes, are the outliers and leverage points valid?
4
Researcher A suggests the transformation X' = 𝐗 𝟐 and researcher B suggests the transformation Y' = √𝐘 .
Which one will you choose and why?