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Math Help - Function/Regression Question

  1. #1
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    Function/Regression Question

    Use the data in the table to model the population of the world in to 20th century by a cubic function. Then use your model to estimate the population in the year 1925.

    Year Population (millions)
    1900 1650
    1910 1750
    1920 1860
    1930 2070
    1940 2300
    1950 2520
    1960 3020
    1970 3700
    1980 4450
    1990 5300
    1996 5770


    Using a graphing calculator to plot the points and do a cubic regression, I get the equation y=.00233x3 - 13.065x2 + 24463.108x - 15265793.873, which matches with the answer in the back of the book. But when I plug in 1925 for x into the equation, I get 32,352.933, which doesn't make since when looking at the table. The answer in the back of the book is 1922 million, which makes more sense because this is a suitable answer since it is between 1750 and 1860 (the values that correspond to years 1910 and 1920). I'm probably missing something very simple. How do I get 1922 million?
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  2. #2
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    Re: Function/Regression Question

    Are you using the actual year number as your x values, or are you say assigning 1900 to be "year 0" and all other years as the number of years after this?
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  3. #3
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    Re: Function/Regression Question

    I am using the actual year number as my x values.
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    Re: Function/Regression Question

    Quote Originally Posted by cdbowman42 View Post
    Use the data in the table to model the population of the world in to 20th century by a cubic function. Then use your model to estimate the population in the year 1925.

    Year Population (millions)
    1900 1650
    1910 1750
    1920 1860
    1930 2070
    1940 2300
    1950 2520
    1960 3020
    1970 3700
    1980 4450
    1990 5300
    1996 5770


    Using a graphing calculator to plot the points and do a cubic regression, I get the equation y=.00233x3 - 13.065x2 + 24463.108x - 15265793.873, which matches with the answer in the back of the book. But when I plug in 1925 for x into the equation, I get 32,352.933, which doesn't make since when looking at the table. The answer in the back of the book is 1922 million, which makes more sense because this is a suitable answer since it is between 1750 and 1860 (the values that correspond to years 1910 and 1920). I'm probably missing something very simple. How do I get 1922 million?
    You do not say what the exact wording or purpose of this exercise is. Using Excel, I get parameters of 0.0023257, - 13.0649, 24,463.108, and -15,265,793.873 and a population of approximately 1922.46 million for 1925, which rounds to the book's solution.

    This model, however, is very sensitive to small errors in the estimated coefficients when x = 1925. Moreover, the model has no theoretical justification: populations usually demonstrate exponential growth, not growth subject to a cubic. Finally, the model is historically absurd: the decades from 1910-1920, 1930-1940, and 1940-1950 do not come from the same "population" as the other intervals. Perhaps the purpose of the exercise is to show that regression analysis requires judgment, not unthinking number crunching. Without seeing the wording of the problem or how it is used in class, I do not know. I suspect, most probably, that you made an arithmetical error (easy enough to do with so many meaningless digits) when applying the model, or, possibly, that a super-sensitive model gave a silly answer when you approximated the parameters to eliminate obviously meaningless digits.
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