Eruptions of the Old Faithful geyser in Yellowstone National Park typically last from 1.5 to 5 minutes. Between eruptions are dormant periods, which typically last from 50 to 100 minutes. A dormant period can also be thought of as the waiting time between eruptions. The durations in minutes for 60 consecutive dormant periods are given in the following table. It is desired to predict the length of a dormant period from the length of the dormant period immediately preceding it. To express this in symbols, denote the sequence of dormant periods T1 ,..., T60 . It is desired to predict Ti +1 from Ti .
|
i
|
Ti
|
i
|
Ti
|
i
|
Ti
|
i
|
Ti
|
i
|
Ti
|
i
|
Ti
|
|
1
|
80
|
11
|
56
|
21
|
82
|
31
|
88
|
41
|
72
|
51
|
67
|
|
2
|
84
|
12
|
80
|
22
|
51
|
32
|
51
|
42
|
75
|
52
|
81
|
|
3
|
50
|
13
|
69
|
23
|
76
|
33
|
80
|
43
|
75
|
53
|
76
|
|
4
|
93
|
14
|
57
|
24
|
82
|
34
|
49
|
44
|
66
|
54
|
83
|
|
5
|
55
|
15
|
90
|
25
|
84
|
35
|
82
|
45
|
84
|
55
|
76
|
|
6
|
76
|
16
|
42
|
26
|
53
|
36
|
75
|
46
|
70
|
56
|
55
|
|
7
|
58
|
17
|
91
|
27
|
86
|
37
|
73
|
47
|
79
|
57
|
73
|
|
8
|
74
|
18
|
51
|
28
|
51
|
38
|
67
|
48
|
60
|
58
|
56
|
|
9
|
75
|
19
|
79
|
29
|
85
|
39
|
68
|
49
|
86
|
59
|
83
|
|
10
|
80
|
20
|
53
|
30
|
45
|
40
|
86
|
50
|
71
|
60
|
57
|
a. Construct a scatterplot of the points (Ti , Ti +1 ), for i = 1,..., 59.
b. Compute the least-squares line for predicting Ti +1 from Ti . (Hint: The values of the independent variable (x ) are
T1 ,..., T59 , and the values of the dependent variable (y) are T2 ,..., T60 .)
c. Find a 95% confidence interval for the slope β1 .
d. If the waiting time before the last eruption was 70 minutes, what is the predicted waiting time before the next eruption?
e. Find a 98% confidence interval for the mean waiting time before the next eruption when the time before the last eruption was 70 minutes.
f. Find a 99% prediction interval for the waiting time before the next eruption, if the time before the last eruption was 70 minutes.