ENGRI 1270 worked example for subst curve November 20 2023

ENGRI 1270 Worked Example November 20, 2023 As you may know, recorded music was purchased in the form of long-playing vinyl records and audio cassettes in the 1970’s. In the 1980’s, compact discs were introduced. Here is the fraction of album purchases in CD format [as opposed to vinyl LP’s and cassette tapes] over time. Year Fraction in CD format (%) 1986 10.0 1988 23.0 1990 38.7 1992 47.7 1994 65.6 a) Using this data, find the constants for the Fisher-Pry curve that best predicts future CD market share. b) Using your equation, find the prediction for 2002 CD market share. Steps: 1. Assign a smaller value to the year parameter; let’s say that 1986 means t = 1, 1988 has t = 3, etc. The year 2002 will be t = 17. 2. Express the percentage as a decimal number. t f 1 0.100 3 0.230 5 0.387 7 0.477 9 0.656 3. Calculate f/(1-f) and take the natural logarithm of that quantity. t f f/(1 – f ) ln[f/(1 – f ) 1 0.100 0.111111 -2.19722 3 0.230 0.298701 -1.20831 5 0.387 0.631321 -0.45994 7 0.477 0.912046 -0.09206 9 0.656 1.906977 0.645519 4. We want to find the slope and intercept of a best fit line through the ordered pairs of points with ln[f/(1 – f )] on the ordinate versus t on the abscissa. You may wish to plot the ordered pairs first, these look like:

5. As you can see, the data points do NOT lie exactly along a straight line, but there is definitely a linear tendency and we can fit a straight line to these points. To get the best fit line, we use the linear regression function on any calculator. Be sure to enter the data in the correct order (which depends on the particular model of calculator you use!). My calculator requires the data to be entered as ordered pairs: (1, -2.19722), (3, -1.20831), (5, -0.45994), (7, -0.09206), (9, 0.645519) 6. Write down the results from the linear regression on your calculator. Slope = 0.340087 Intercept = - 2.36284 7. The Slope value is simply b, so that one is all set and ready to use. We know the intercept has to be manipulated: -ln a = intercept. a = e -intercept = e 2.36284 = 10.62105 8. Use these values of a and b along with the desired value of t to find the fraction using the new technology at any time. We have been asked to find the fraction at t = 17. ( ) ( )( ) bt 0.340087 17 1 1 f 0.968282 1 ae 1 10.62105 e − − = = = + + We see that 96.8282% of recordings are expected to be in CD format in 2002. The actual number was slightly less than this amount, but the forecast appears to have been a good one.

4. Find k and b intercept 1.020844479 k slope 0.204066 b e e 2.775538 = − = = = = 5. Plug these values and t = 17 into the Gompertz equation. ( ) ( )( )   intercept 1.020844479 f exp( bexp( kt)) k slope 0.204066 b e e 2.775538 t 17 f exp 2.775538 exp 0.204066 17 f 91.72% = − − = − = = = = = = − −     = The Fisher-Pry is a better prediction in this case.