USING FUNCTIONS AS TIME MACHINES The Consumer Price Index (CPI) is a measure of the perceived "value" of a U.S. dollar. Data is collected by the U.S. Department of Labor. A standard "basket" of goods and services are "purchased" and the total cost is compared to the cost at a reference time. In the data below, the reference is 1982 through 1984 = 100, since the average during the years 1982-1984 is 100. This also means that the CPI is in reference to the average buying power of $1 in the years 1982-1984. For example, the CPI is 118.3 in 1988. This means that it cost $1.18 3/10 in 1988 to buy the same goods and services that cost $1, on the average, in 1982-1984. Similarly, since the CPI was 31.0 in 1964, the goods and services that could be obtained for $1 in 1982-1984 cost only $0.31 in 1964. Inflation is defined as the percent change in the CPI from year to year. YEAR CPI YEAR CPI 1960 29.6 1978 65.2 1961 29.9 1979 72.6 1962 30.2 1980 82.4 1963 30.6 1981 90.9 1964 31.0 1982 96.5 1965 31.5 1983 99.6 1966 32.4 1984 103.9 1967 33.4 1985 107.6 1968 34.8 1986 109.6 1969 36.7 1987 113.6 1970 38.8 1988 118.3 1971 40.5 1989 124.0 1972 41.8 1990 130.7 1973 44.4 1991 136.2 1974 49.3 1992 140.3 1975 53.8 1993 144.5 1976 56.9 1994 148.4 1977 60.6 1995(March) 151.4 1. Plot the above data accurately on a coordinate system, labeling each axis appropriately as CPI (x) and TIME (y). You must decide how you will measure time before you actually mark your axes. Please use graph paper for accuracy. 2. Try to draw, in color and as accurately as possible by hand, a "line of best fit" for your data points. This would be a straight line that might actually go through some of the points (but might not) and lies on the graph in such a way that there are approximately an equal number of points on either side of it. 3. Approximate a point on each "end" of the graph that actually lies on the line. Use these 2 points to write the slope-intercept form of the line you drew. 4. To get a more accurate equation, use the TI-85. Push the STAT button...choose F2, the EDIT command. Type CPI, push ENTER, then type TIME, push ENTER. Type your first CPI, push ENTER, then type your associated time, push ENTER. Repeat until you have entered all of your pairs. When finished, push EXIT button. Now choose F1, CALC, to calculate the results. Push ENTER twice. Choose F2, LINR. You should have a screen that displays a,b,corr and n. If corr is a number close to 1 or -1, you have a very accurate function for your data. If corr equals 1 or -1, all data points are on the graph of the function. If corr is positive, the function is increasing and if corr is negative the function is decreasing. The letters a and b are the constants in the function: T(x)= a + bx, where x is the CPI and T(x) represents the year (or number of years since 1960) when the CPI has a value of x. This function is now a model for this situation. Please write out the function, as above. 5a. Go back to the STAT menu and choose F1, CALC, to calculate the results. Push ENTER twice. Choose F3, LNR, this time. The screen looks the same as in number 4, but this time, the letters a and b represent constants in the function: T(x) = a + b*ln(x), x > 0, where x is the CPI and T(x) represents the year (or number of years since 1960) when the CPI has a value of x. Please write out the function, as above. This function is a better model for this situation. How do you know that this is true? b. Push EXIT once and then choose DRAW, F3. Choose F2, SCAT, to show all of your data points and then choose F4, DRREG, to draw the function you just created. Our model function is not a perfect predictor of CPI. By looking at your graph or the graph on the TI-85, decide the periods of time that our function here overestimates the CPI and when it underestimates the CPI. Remember that TIME is on the y-axis, so you might want to turn the graph sideways. 6. Let's use the function in number 5 and the FCST, F4, under STAT to predict the future by answering the following questions: a. When should the CPI be 200? b. What should the CPI be 10 years from now? c. When should the CPI be 1000? 7. If the CPI is 300, that means that prices have tripled since the years 1982-1984. Decide what the CPI should be so that the dollar we use today is, for all practical purposes, worthless. Explain in a sentence or two why your group chose that particular value. Then, use our future-predicting function to predict what year that CPI will be reached.