Non-Linear Data Homework

Year / # of Subscribers
(millions)
0 / 5.3
1 / 7.6
2 / 11.0
3 / 16.0
4 / 24.1
5 / 33.8
6 / 43.8

1. The following table shows the growth of cell phone subscribers from 1990 to 1996 (from YMS test bank). Note year 0 = 1990.

  1. Sketch a scatterplot of the data with its least squares regression line.
  2. Sketch and interpret the residual plot for the linear model.
  3. Sketch a scatterplot of log(subscribers) vs. year with its least squares line.
  4. Sketch and interpret the residual plot for the exponential model.
  5. Which model is better? Explain.
  6. Predict the number of subscribers in 1997 and 2003 using both models. Are you confident in your predictions? Explain.

Year / # Transistors
1 / 2250
2 / 2500
4 / 5000
8 / 29,000
12 / 120,000
15 / 275,000
19 / 1,180,000
23 / 3,100,000
27 / 7,500,000
29 / 24,000,000
30 / 42,000,000

2. The following table shows the number of transistors in an Intel microchip over the period from 1971 to 2000. For ease of calculation, large numbers are sometimes rescaled to smaller numbers. In this case, 1970 will be called year 0, 1971 = year 1, etc. (YMS problem 4.7).

  1. Sketch a scatterplot of the data with its least squares regression line.
  2. Sketch and interpret the residual plot for the linear model.
  3. Sketch a scatterplot of log(number) vs. year with its least squares line.
  4. Sketch and interpret the residual plot for the exponential model.
  5. Which model is better? Explain.
  6. Using the both models, predict the number of transistors in the year 2003. Are you confident in either prediction?

3. The following data show the relationship between degree of exposure to 242Cm alpha particles (x) and the percentage of exposed cells without aberrations (y) (from POD problem 5.43).

x / .106 / .193 / .511 / .527 / 1.08 / 1.62 / 1.73 / 2.36 / 2.72 / 3.12 / 3.88 / 4.18
y / 98 / 95 / 87 / 85 / 75 / 72 / 64 / 55 / 44 / 41 / 37 / 40
  1. Sketch a scatterplot of the data with its least squares regression line.
  2. Sketch and interpret the residual plot for the linear model.
  3. Sketch a scatterplot of log(percentage) vs. degree with its least squares regression line.
  4. Sketch and interpret the residual plot for the exponential model.
  5. Which model is better? Explain.
  6. Using the both models, predict the percentage of exposed cells without aberrations for a degree of exposure of x = 3. Are you confident in your predictions?

4. An experiment was conducted in which a colony of bacteria was exposed to X-rays. The number of surviving bacteria (in hundreds) is shown for time periods from 1 to 15 minutes (YMS problem 4.77).

Time / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12 / 13 / 14 / 15
Number / 355 / 211 / 197 / 166 / 142 / 106 / 104 / 60 / 56 / 38 / 36 / 32 / 21 / 19 / 15
  1. Sketch a scatterplot of the data with its least squares regression line.
  2. Sketch and interpret the residual plot for the linear model.
  3. Sketch a scatterplot of log(number) vs. time with its least squares line.
  4. Sketch and interpret the residual plot for the exponential model.
  5. Which model is better? Explain.
  6. Predict the original number of bacteria in the colony using both models. Which prediction is likely to be closer to the true value?

Amount of
Supplement / Increase in
Muscle mass
500 / 3.6
500 / 4.3
500 / 5.0
1000 / 5.5
1000 / 6.6
1000 / 7.5
2000 / 7.9
2000 / 8.9
2000 / 9.8
3000 / 10.1
3000 / 10.9
3000 / 11.6

5. In an experiment to test if a new nutritional supplement helps increase muscle mass, 12 similar volunteers were recruited and assigned to four different levels of the supplement (in mg/day). After 4 weeks of using the supplement and following a specific weightlifting routine, researchers measured the increase in the volunteers’ muscle mass (in pounds).

  1. Sketch a scatterplot of the data with its least squares regression line.
  2. Sketch and interpret the residual plot for the linear model.
  3. Sketch a scatterplot of log(mass) vs. amount with its least squares line.
  4. Sketch and interpret the residual plot for the exponential model.
  5. Sketch a scatterplot of log(mass) vs. log(amount) with its least squares line.
  6. Sketch and interpret the residual plot for the power model.
  7. Which of the 3 models is best? Explain.
  8. Using all three models, predict the increase in muscle mass if a volunteer was assigned to take 2500 mg/day.

Distance / Intensity
1.0 / .2965
1.1 / .2522
1.2 / .2055
1.3 / .1746
1.4 / .1534
1.5 / .1352
1.6 / .1145
1.7 / .1024
1.8 / .0923
1.9 / .0832
2.0 / .0734

6. In a physics class, the intensity (in candelas) of a 100 watt light bulb was measured by a sensing device at various distances (in meters) from the light source and the following data were collected (YMS problem 4.72)

  1. Sketch a scatterplot of the data with its least squares regression line.
  2. Sketch and interpret the residual plot for the linear model.
  3. Sketch a scatterplot of log(intensity) vs. distance with its least squares line.
  4. Sketch and interpret the residual plot for the exponential model.
  5. Sketch a scatterplot of log(intensity) vs. log(distance) with its least squares line.
  6. Sketch and interpret the residual plot for the power model.
  7. Which of the 3 models is best? Explain.
  8. Using all three models, predict the intensity of a 100 watt bulb 2.1 meters away. Do you have confidence in any of the predictions?

Stimulus
(decibels) / Response
(loudness)
30 / .2
50 / 1
60 / 3
70 / 5
75 / 8.5
80 / 10
85 / 14
90 / 20
95 / 29
100 / 43

7. A researcher asked a number of subjects to compare notes of various decibel (sound) levels against a standard (set at 80 decibels) and to assign them a loudness rating. Subjects were told to assign to the standard note of 80 decibels a loudness rating of 10. The average response for various decibel levels are shown in the table below (from Statistics: Concepts and Methods):

  1. Sketch a scatterplot of the data with its least squares regression line.
  2. Sketch and interpret the residual plot for the linear model.
  3. Sketch a scatterplot of log(loudness) vs. stimulus with its least squares line.
  4. Sketch and interpret the residual plot for the exponential model.
  5. Sketch a scatterplot of log(loudness) vs. log(stimulus) with its least squares line.
  6. Sketch and interpret the residual plot for the power model.
  7. Which of the 3 models is best? Explain.
  8. Using all three models, predict the loudness of a 40 decibel stimulus. Do you have confidence in any of the predictions?