Ap Statistics
4.1 Worksheet Name:
The number of motor vehicles registered in the U.S. has grown as follows:
YearVehiclesYearVehicles
1940 32.41965 90.4
1945 31.01970 108.4
1950 49.21975 132.9
1955 62.71980 155.8
1960 73.91985 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.
Clearly, the point (1945,1.49) is an outlier. Can you suggest an explanation?
3. 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.
4. Determine the exponential equation (model) for the original dataset. (Form: y = c•10^ kx )
The following table shows the federal debt for the years 1980 through 1991.
Federal Debt
Year(in trillions)
19800.909
19810.994
19821.1
19831.4
19841.6
19851.8
19862.1
19872.3
19882.6
19892.9
19903.2
19913.6
5.Construct a scatterplot on the grid provided. Perform an appropriate test to show that the data are (approximately) exponential.
6.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.
7.Now transform your linear equation back to obtain a model for the original federal debt data. (It should be in the form y = c•10 kx ) Write the equation for this model.
8.Compare your model’s predictions for 1990 and 1991 to the actual federal debt.
9.Use your model to predict the national debt in the year 2000.
10.What is it called when you predict for a point outside the range of your data?
4.1 Worksheet (extra practice)Name:
Stamp collectors know that the United States and other countries have issued a very large number of different stamps over the years. They may not realize, however, that postage stamp production in the United States has been increasing at an exponential rate. The following table shows the cumulative number of regular and commemorative U.S. postage stamp issues by 10-year intervals from 1848-1988:
Number of U.S.
Stamps issued
Year(cumulative)
1848 2
1858 30
1868 88
1878 181
1888 218
1898 293
1908 341
1918 529
1928 647
1938 838
1948 980
1958 1123
1968 1364
1978 1769
1988 2400
1.Perform an appropriate test to show that these data are increasing in an (approximate) exponential manner. Record the appropriate test numbers in the table above.
2.Plot a scatterplot of these data on the grid provided. Label and scale both axes.
3.On your calculator, use a logarithmic transformation to produce a linear scatterplot of log y on x = year.
4. Notice that the first three points do not fit the linear pattern of the rest of the transformed points. Delete the first three data points, and perform least squares regression on the remaining data points. Complete the model by performing the inverse transformation. Write the exponential equation for your model in the space below, and sketch the exponential curve on the graph above.
5.Use your model to predict the cumulative number of U.S. postage stamp issues in the year 2000.
Chapter 4.1 Solutions
Worksheet 4.1
(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 = ( 10 ^–30.2406 )( 10 ^.0164x ).
(5) 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. (6) 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 = –110.6417 + .05586x. (7) Performing the inverse transformation yields = ( 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. (8) (2000) = 11.9888 or about $12 trillion. (9) Extrapolation.
Worksheet 4.1(Extra Practice)
(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.
Revised by Mrs. Poyner1Section 4.1 (Yates)