4.1 Transformations
1.
AP Statistics
Name:
The following data were collect by running an experiment in which 20 dice were tossed. All of the dice that showed 2s and 3s
were removed, and the remaining dice were tossed. Again, all of the dice showing 2s and 3s were removed, and so, on.
Toss Number
Number of Dice Remaining
0
20
1
14
2
8
3
6
4
4
5
2
6
1
7
1
a.
Make a scatterplot of the data. Use Zoom #9 STAT as you window. Graph on the plot below.
b.
What type of model do you think this data represents?
c.
What is the correlation coefficient (r) and the coefficient of determination for
this relationship if it were model by a linear least-squares regression line?
Explain what r and r2 means for this situation.
d.
How can you validate that an exponential model is a good model for this data?
e.
Validate that an exponential is a good model to use. Graph x and log y on the graph below.
f.
What is (r) and (r-squared) for the plot of x and log y you found in (e)?
(Remember to do linear regression on L1 and L3)
g.
What is your exponential regression model?
8
1
2.
Earthquakes are among the most damaging kinds of natural disasters. The size of an earthquake is generally reported as a rating
on the Richter scale-usually a number between 1 and 9. That Richter scale rating indicates the energy released by the shaking of
the ground and the height of the shock waves recorded on seismographs.
The data in the following table show Richter scale ratings and amounts of energy released for six earthquakes.
Earthquake Location
San Francisco, CA, 1906
Yugoslavia, 1963
Alaska, 1964
Peru, 1970
Italy, 1976
Loma Prieta, CA, 1989
a)
Richter
Scale
Rating
8.25
6.0
8.6
7.8
Energy
(in sextillion
ergs)
1500
0.63
5000
320
6.5
7.1
3.5
28
Use your calculator to make a scatter plot for this data. Explain why an exponential model would be a good model to use.
b) Write an exponential regression model for this data.
c)
How confident would you be with predicting the Energy if the Richter Scale Rating was a 6.3? Why?
d) Use the model to estimate energy released by earthquakes listed in the following chart.
Earthquake Location
Quetta, India, 1906
Richter Scale
Rating
7.5
Kwanto, Japan, 1923
8.2
Chillan, Chile, 1939
7.75
Agadir, Morocco, 1960
5.9
Iran, 1968
7.4
Tangshan, China, 1976
7.6
Northridge, CA, 1994
Kobe, Japan, 1995
6.7
7.2
Energy
(in sextillion ergs)
3.
During the 1980s the population of a certain city went from 100,000 to 205,000. Populations by year are listed in the table below.
Year
Population in thousands
1980
100
1981
108
1982
117
1983
127
1986
163
1987
175
1988
190
1989
205
a.
Find the least squares regression line for this data. Write the equation and the r & r—squared value below.
b.
Would a linear regression line be a good model for this data? Explain.
c.
Since a linear model would not be a good model, which transformation will you use to straighten the data?
d.
Write the exponential equation for this data.
e.
Find the population in 1984 and 1985. How confident are you in your answer?
f.
The actual population in 1984 and 1985 was 138,000 and 149,000. How good was the model in predicting the population in
1984 and 1985? Did you expect this? If so, why?
4.
Ryan drops a ball from various heights and records the time, x, that it take for the ball to hit the ground, using a motion detector.
He obtains the data displayed in the table.
Time
Distance
1.528
11.46
2.015
19.99
a. Look at the distance data. Is a linear model a good model to use? Explain. How COULD you
3.852
72.41
determine if a linear model is a good model to use?
4.154
84.45
4.625
104.23
b.
Look at the distance data. Would an exponential be a good model to use? Explain. How COULD you determine if an
exponential model is a good model to use?
c.
What transformation will you use to straighten the data?
d.
Find the equation of the power regression model.
e.
Predict the distance a ball would have to fall in order to take 4.2 seconds to hit the ground.