L1.10: Exercises

Part I

Reproduce the results in Examples 1–12.

Part II—Work the assigned problems

[13] A bank balance earning a constant rate of compound interest has these values: $1550 after 5 years, $2002 after 10 years, $2585 after 15 years, $3339 after 20 years, and $4313 after 25 years. What was the original deposit amount (that is, the balance after 0 years), and what annual interest rate was applied?

[14] The activity of a radioactive substance is measured on the same day each year for several years, with these results: 5.7 Curies after 1 year, 3.8 Curies after 2 year, 2.6 Curies after 3 years, and 1.7 Curies after 4 years. What is the decay rate of the substance? What will the activity be after 10 years?

Problems 15–24 have the same instructions, applied to different datasets. Copy and paste the datasets from the course web site copy of this topic into the Models.xls spreadsheet, rather than retyping them.

For each of the datasets listed below

  1. Display the dataset and visually determine which of the models discussed in this topic is most suitable for this data.
  2. Identify which points, if any, are outliers for this dataset.
  3. Fit an appropriate model to the dataset, omitting the outliers (if any are present).
  4. Report the best-fit model parameters and their standard deviation for this data.
[15] Dataset A

x y
5 457.4
10 250.9
15 138.7
20 76.2
25 41.4
30 22.9
35 12.6
40 78.0
45 4.5
50 1.8
55 1.4
60 0.8
[16] Dataset B

x y
0 172
1 195
2 216
3 230
4 244
5 256
6 261
7 266
8 264
9 262
10 255
11 247
[17] Dataset C

x y
0 314.27
0.5 297.66
1 282.01
1.5 267.46
2 249.19
2.5 235.10
3 218.96
3.5 20.06
4 184.62
4.5 166.80
5 145.85
5.5 131.63
[18] Dataset D

x y
1 239.7
2 296.6
3 386.6
4 469.9
5 597.6
6 777.3
7 952.2
8 1180.0
9 1424.4
10 1682.6
11 1980.3
12 2309.7
[19] Dataset E

x y
1992 45,619
1993 49,529
1994 53,405
1995 57,228
1996 60,877
1997 65,003
1998 68,849
1999 72,399
2000 76,529
2001 80,448
2002 84,030
2003 88,027
[20] Dataset F

x y
0 -68
1 -66.1
2 -62.6
3 -56.3
4 -47.3
5 -35.4
6 -23.4
7 -1.0
8 11.4
9 34.3
10 61.3
11 91.8
12 119.3
13 151.8
14 188.5
15 225.8
16 266.4
17 309.1
18 352.1
19 402.4
20 451.2
[21] Dataset G

x y
0 10.65
1 8.46
2 7.10
3 5.60
4 4.74
5 3.90
6 3.09
7 2.62
8 2.00
9 1.55
10 1.47
11 0.98
12 0.76
13 0.97
14 0.62
15 0.49
16 0.42
17 0.29
18 0.27
19 0.21
20 0.20
[22] Dataset H

x y
5.7 740.79
20.0 722.19
44.4 690.51
61.0 668.94
76.8 648.33
117.2 595.89
125.9 684.50
133.7 574.36
151.9 550.79
176.8 518.33
191.5 499.26
205.8 480.61
225.0 455.74
237.9 438.93
250.4 422.71
276.2 389.13
298.2 360.53
321.4 330.37
343.9 201.12
348.7 294.86
379.0 255.51
[23] Dataset I

x y
0 50.9
1 158.9
2 63.9
3 71.5
4 77.7
5 83.6
6 87.9
7 92.9
8 94.4
9 94.1
10 98.2
11 99.3
12 100.1
13 98.4
14 100
15 96.5
16 92.8
17 90.6
18 88
19 83.4
20 77.5
[24] Dataset J

x y
232.27 448.127
87.53 634.918
307.27 346.181
98.05 620.371
312.34 342.395
277.44 387.947
462.19 147.666
211.58 471.990
145.38 558.238
309.27 346.403
449.25 164.451
196.22 491.841
335.40 312.652
187.02 505.495
131.27 577.226
354.55 287.090
336.17 310.246
369.39 266.686
124.17 586.563
214.03 467.349
386.05 246.012

[25] Scientists have found that the total energy requirements of animals increase somewhat more slowly than body size. For example, a 1.2-pound mongoose requires 47 kilocalories per day, a 10-pound fox requires 240, a 22-pound bobcat requires 440, a 100-pound wolf requires 1350, a 300-pound lion requires 3100, a 400-pound tiger requires 3850, and a 700-pound polar bear requires 5900.

  1. What are the best-fit parameters to this data for a “power” model? [The general formula for a power model is y = a * x^b.]
  2. What is the standard deviation of the data from the best-fit model?
  3. Does this data support the idea that a power model is appropriate for predicting the energy requirements of animals?
  4. What daily energy requirement can be expected for a 45-pound lynx?

[26] The frequency of earthquakes varies by their size, with stronger ones being less frequent. In a recent one-year reporting period, the number of earthquakes detected at a particular facility was: 302,417 magnitude-2 quakes, 36,288 magnitude-3 quakes, 4,354 magnitude-4 quakes, 525 magnitude-5 quakes, and 60 magnitude-6 quakes.

  1. What are the best-fit parameters to this data for an exponential model, where the magnitude is the input parameter and the earthquake count is the output variable?
  2. What is the standard deviation of the data from the best-fit model?
  3. Does the data support the idea that this relationship is exponential?
  4. How many magnitude-7 earthquakes does this model predict this facility will detect each year?

[27] For Dataset C, find the inverse model (that is, the model when the x and y columns are swapped).

[28] For Dataset J, find the inverse model (that is, the model when the x and y columns are swapped).

[29] For Dataset G, use Solver to find the best-fit parameters if the spreadsheet is modified to minimize the relative standard deviation. Compare these parameters to those found in Exercise 21.

 

Exercise 30

Dataset K

x y
0 0.192
0.25 0.171
0.5 0.152
0.75 0.140
1 0.129
1.25 0.117
1.5 0.112
1.75 0.104
2 0.098
2.25 0.088
2.5 0.087
2.75 0.081
3 0.075
3.25 0.073
3.5 0.069
3.75 0.068
4 0.061
4.25 0.061
4.5 0.058
4.75 0.057
5 0.055
 

Exercise 31

Dataset L

x y
10 263
20 378
30 453
40 525
50 585
60 646
70 693
80 744
90 789
100 827
110 871
120 908
130 943
140 985
150 1013
160 1052
170 1081

Exercise 32

Dataset M

x y
0 8
2 193
4 364
6 529
8 657
10 722
12 725
14 678
16 531
18 426
20 235
22 61
24 -162
26 -335
28 -467
30 -637
32 -693
34 -721
36 -670
38 -570
40 -461
42 -303
44 -61
46 98
48 300
50 468
52 579
54 688
56 735
58 713
60 620
62 498
64 325
66 134
68 -57
70 -248
72 -437
74 -574
76 -683
78 -724
80 -743
82 -645
84 -513
86 -345
88 -182
90 13
92 216
94 405
96 540
98 660
 

Exercise 33

Dataset N

x y
-3.0 0
-2.8 0
-2.6 0
-2.4 1
-2.2 2
-2.0 4
-1.8 11
-1.6 17
-1.4 35
-1.2 60
-1.0 97
-0.8 94
-0.6 187
-0.4 205
-0.2 236
0.0 237
0.2 228
0.4 208
0.6 182
0.8 118
1.0 82
1.2 51
1.4 25
1.6 15
1.8 9
2.0 1
2.2 2
2.4 0
2.6 0
2.8 0
3.0 0
 

Exercise 34

Dataset O

x y
-7 0.0
-6 0.2
-5 0.9
-4 1.2
-3 3.0
-2 5.0
-1 8.1
0 12.2
1 16.6
2 19.7
3 21.6
4 23.2
5 23.8
6 24.2
7 24.3

Exercise 30-34 Instructions

The formulas supplied below (in both algebraic form and as the spreadsheet formula for C3) will fit the corresponding dataset well as soon as the best settings are found for the parameters a and b.

For each formula supplied, make a worksheet that uses it as a model.   Then put the specified dataset into the worksheet and use the Solver tool to find the a and b values that fit the dataset best.

For Dataset K, use formula

[latex]y=\frac{1}{(a+bx)}[/latex]

=1/($G$3+$G$4*A3)

[Start with G3=5 & G4=1]

 

For Dataset L, use formula

[latex]y=a\cdot{{x}^{b}}[/latex]

=$G$3*A3^$G$4

[Start with G3=1 & G4=1]

 

For Dataset M, use formula

[latex]y=a\cdot\sin(b\cdotx)[/latex]

=$G$3*SIN($G$4*A3)

[Start with G3=1000 & G4=1]

 

For Dataset N, use formula

[latex]y=a\cdot{{b}^{-{{x}^{2}}}}[/latex]

=$G$3*($G$4^-(A3^2))

[Start with G3=100 & G4=2]

 

For Dataset O, use formula

[latex]y=\frac{a}{1+{{b}^{x}}}[/latex]

=$G$3/(1+$G$4^A3)

[Start with G3=10 & G4=1]

 

[Optional: for each model, choose names for the a and b parameters that are suggestive of the effect that parameter has on the model.]