Name
___ Period______
T. Iverson Section 4.1 Worksheet
The number of motor vehicles registered (in millions) in the U.S. has grown as follows:
Year Vehicles
Year Vehicles
1940
32.4
1965
90.4
1945
31.0
1970 108.4
1950
49.2
1975 132.9
1955
62.7
1980 155.8
1960
73.9
1985 171.7
1. Plot the number of vehicles against time.
What kind of growth does this exhibit?
2. Use logs to transform the data into a
linear association, and plot the transformed data.
3. Clearly, the point (1945, 1.49) is an outlier. Can you suggest an explanation?
4. Delete the outlier (x=1945) and use the remaining points to find the LRSL equation of Log Y on X. Write the
equation below, and draw this line on your second (linear) graph. (Identify at least two points on your line)
5. Determine the exponential equation (model) for the modified dataset. Predict the number of vehicles registered in
1990.
The following table shows the federal debt for the years 1980 through 1991.
Federal Debt
Year
(in trillions)_________________________
1980
0.909
1981
0.994
1982
1.1
1983
1.4
1984
1.6
1985
1.8
1986
2.1
1987
2.3
1988
2.6
1989
2.9
1990
3.2
1991
3.6
6.
Construct a scatterplot on the grid provided, let x equal the number of years since 1979. Perform an appropriate
test to show that the data are approximately exponential. (Look at the ratio of consecutive federal debt amounts).
7.
Calculate the logarithms of the y-values and extend the table above to show the transformed data (just write the
first four logs). Then perform least square regression on the transformed data. Write the LSRL equation for the
transformed data:
What is the correlation coefficient?
Is this correlation between YEAR and FEDERAL DEBT? Explain briefly.
8.
Now transform your linear equation back to obtain a model for the original federal debt data. (It should be in the
form y = a•b x ) Write the equation for this model. Show your work.
9.
Compare your model’s predictions for 1990 and 1991 to the actual federal debt.
10. Use your model to predict the national debt in the year 2000.
Chapter 4 Solutions
Quiz 4.1A
(1) See the first screen below. YEAR is explanatory, NO. OF VEHICLES is response. The growth is
exponential: the common factor is ~1.20. (2) The second screen shows LOG(NO. OF VEHICLES)
vs. YEAR. Except for the point (1945, 1.49), the pattern is very linear. An explanation for this outlier
is World War II. Men and women were overseas, involved in the war effort. Metal was going to make
planes and tanks, not cars. (3) Deleting the outlier and regressing LOGY on X yields the linear
equation LOGY = –30.2406 + .0164X. See Plot. (4) Performing the inverse transformation gives y =
( 10 ^–30.2406 )( 10 ^.0164x ).
Quiz 4.1B
(1) The first screen below shows the scatterplot. This growth data is exponential: there’s a common
factor: 1.09, 1.09, 1.27, 1.14, 1.13, 1.17, etc. (2) The first four logs are: –.0414, –.0026, .0414,
.1461. The transformed data are shown in the second screen. The LSRL equation for the transformed
data is: log yˆ = –110.6417 + .05586x. (4) Performing the inverse transformation yields yˆ = ( 10 ^–
110.6417 )( 10 ^.05586x ). For 1990, the observed debt is 3.2 (trillion), and the predicted is 3.3126.
For 1991, the observed debt is 3.6 and the predicted is 3.767. (6) yˆ (2000) = 11.9888 or about $12
trillion. (7) Extrapolation.
Quiz 4.1C
(1) The ratios of y-values to previous y-values are: 15, 2.9, 2.1, 1.2,1.3, 1.2, 1.6, 1.2, 1.3, 1.2, 1.1, 1.2,
1.3, 1.4. Since the ratios are approximately constant, growth is approximately exponential. (2) See
scatterplot below. Note that we coded the years (using 48 for 1948, for example). (3) See regression
line of transformed points. (4) The least squares equation for the reduced data set is log (Stamps) =
1.4863 + 0.01004 (Year). Back transforming gives us
Stamps = 30.6428 (10^(0.01Year)). (5) Our exponential model predicts Stamps(200)  3119 stamps in
the year 2000.