(Chapter Opener Revisited) In the discussion that introduced this chapter, we looked at a plot that displayed the growth of the Internet over the years from 1993 to 1999. The plot was based on data from the Internet Software Consortium giving the number of Internet hosts, which is displayed in the following table.
Calendar year | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 |
|
t (years after 1993) | 0 | 1 | 2 | 3 | 5 | 5 | 6 |
|
H (millions) | 1.3 | 2.2 | 4.9 | 9.5 | 16.1 | 29.7 | 43.2 |
|
a. Find a natural growth function H1(t) that models the number of Internet hosts as a function of years after 1993 and agrees with the data given for 1993 and 1998.
b. At what average annual rate was the number of Internet hosts growing over this time period? What is the sum of squares of errors SSE1 in the model H1(t) considering the given data for the years through 1993 through 1998 but ignoring the data for 1999?
c. Use your model to predict the number of Internet hosts in 2004. What was the error in your approximation if the actual value was 317.6 million?
d. Your prediction is so much larger than the actual value because the number of Internet hosts did not continue to grow at the rate you found in part (b). The data in the following table give the number of Internet hosts for the years 1998 to 2002.
Calendar year
1998
1999
2000
2001
2002
t (year after 1993)
5
6
7
8
9
H (millions)
29.7
43.2
72.4
109.6
147.3
Source: ISC Internet Domain Survey.
Find a natural growth function H2(t) that models the number of Internet hosts as a function of years alter 1993 and agrees with the data given for 1998 and 2002.
e. At what average annual rate was the number of Internet hosts growing over this time period? What is the sum of squares of errors SSE2 in the model H2(t) considering the given data for the years through 1998 through 2002?
f. Use your model to predict the number of Internet hosts in 2004. Find the error in your approximation, given that the actual value for 2004 was 317.6 million.
g. The two exponential functions H1(t) and H2(t) in parts (a) and (d) agree at the point t = 5 corresponding to 1998. Consider the function H(t) that equals H1(t) for t ≤ 5 but equals H2(t) for t ≥ 5. Can you see why H(t) might be called a “piecewise-exponential model" for the number of Internet hosts from 1993 to 2002? What is the SSE of this model considering the data from 1993 to 2002. Can you find SSE quickly—without doing any more real computation—using your answers to parts (b) and (c)?
h. What is the average error in the piecewise-exponential model H(t) of (g) for the data from 1993 to 2002?
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