Modeling Old Faithful’s Eruptions (TI)
Hot springs that erupt intermittently in a column of hot water and steam are
called geysers. Geysers may erupt in consistent or inconsistent intervals, and
the intervals can last minutes, days, or even years.
Three components are necessary for a geyser to exist: a source of heat and
water, a series of fissures and factures called a plumbing system, and a rock
formation that can withstand the pressure of steam explosions.
One geyser whose eruptions are predictable is named Old Faithful and is
located in Yellowstone National Park in Wyoming.
Some observations about Old Faithful:
•
On the average, eruptions reach heights of 106 to 184 feet and last 1.5 to 5.5 minutes.
•
The interval between eruptions can be as short as 30 minutes or as long as 2 hours.
•
The yearly average for the interval between eruptions has always been between 60 and 79
minutes.
•
Each eruption ejects about 3700 to 8400 gallons of water.
•
The temperature of water at Old Faithful’s vent is about 204°F.
Check out the Yellowstone web site at: http://www.nps.gov/yell/ .
The duration of Old Faithful’s eruptions and the interval between the eruptions is given in the
table below. Let x represent the duration of the geyser eruption in minutes. Let y represent the
interval between geyser eruptions in minutes.
Duration, x
1.80
1.82
1.88
1.90
1.92
1.93
1.98
2.03
2.05
2.13
2.30
2.35
Interval, y
56
58
60
62
60
56
57
60
57
60
57
57
Duration, x
2.37
2.82
3.13
3.27
3.65
3.70
3.78
3.83
3.87
3.88
4.10
4.27
Interval, y
61
73
76
77
77
82
79
85
81
80
89
90
Duration, x
4.30
4.30
4.43
4.43
4.47
4.47
4.53
4.55
4.60
4.60
4.63
Interval, y
84
89
84
89
86
80
89
86
88
92
91
A scatter plot of the data is given below.
Old Faithful Eruptions
100
90
Interval (in minutes)
80
70
60
50
40
30
20
10
0
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
Duration (in minutes)
1.
Have the calculator find the equation of the line of best fit. Also have the calculator find
the R 2 value and the correlation coefficient (R). Record all the results below using 6
decimal places for all constants or coefficients.
Linear Equation: ________________________________________
Slope: ____________________
Y-intercept: ____________________
R 2 value: ____________________
R value: ____________________
2.
Use the table in the TI-83 to find the predicted interval between geyser eruptions (using
the linear regression formula) when the duration of the eruption is 1.50 minutes and 5
minutes. Round to the nearest minute.
1.50 minutes _______________
5 minutes
_______________
Show the use of the equation for one of the calculations above.
3.
Using these two points and the y-intercept, graph the regression line on the scatter plot
above.
5.00
4.
By the correlation coefficient, what is the strength of the relationship between duration
(x, in minutes) and interval (y, in minutes)?
(a) strong negative
(d) weak positive
(b) weak negative
(e) strong positive
(c) no significant relationship
5.
Discuss the meaning of the slope in the context of this data set and the variables involved.
Include specific measurements in your discussion.
6.
In the observations on page 1, it is mentioned that the “practical range” for this function
may be [30, 120], “as short as 30 minutes or as long as 2 hours.” What appears to be the
“practical domain” in minutes for this realistic function?
(a) all real numbers
(b) [106, 184]
(c) [1.50, 5.50]
Helpful linear regression links! —
• http://www.stat.uiuc.edu/~stat100/java/GCApplet/GCAppletFrame.html
• http://lasi.lynchburg.edu/peterson_km/public/old/java_examples/leasts2.htm
This activity is due on ___________________________ .
Do your best! Rise to the challenge! Live and learn!
(d) [ 0, ∞ ]