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Problem 36 Hard Difficulty

The table gives the population of the United States, in millions, for the years 1900-2010. Use a graphing calculator with exponential regression capability to model the US population since 1900. Use the model to estimate the population in 1925 and to predict the population in the year 2020.

Answer

The exponential model is: $y=80.8498 \times e^{0.0126 x}, x$ is the elapsed years from
1900 and $y$ is the population in millions of people.
The estimated population in 1925 is $\approx 11$ millions of people.
The predicted population in 2020 is $\approx 367$ millions of people.

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Video Transcript

in this problem, we're going to use a graphing calculator to compute an exponential regression for the data given in the table. So we go to the stat menu and then into edit, and we type our numbers into list one and list, too. And the numbers I typed into list one represent the number of years since the year 1900. So year zero is 1900. Year 10 is 1910 etcetera. So I didn't type the numbers exactly from the table. I subtracted 1900 from each of them. And then the numbers in column two are the populations. So what we want to do with this is find the exponential regression. And it might be interesting also to look at the scatter plot and see if it looks like exponential growth. So we can go into the stat plot menu. We can turn on plot one when we can go to zoom and go to zoom stat number nine and we can see our scatter plot. Okay, so from here, we want to go ahead and find the regression equation. So we go back to stat over to calculate, and then we go down until we find exponential regression is a bit farther down in the list. There we go. Exponential regression. We pressed. Enter, we're using List one and list, too. We do want to store the regression equation in our Y equals menu. So when you get to this point, you press the variables button, go over to why variables choose function and choose why one and then we can calculate. So if we round these numbers, we have approximately y equals 80.8 times, 1.1 to the X power. Okay, so if you press why equals, you will see that that has now been pasted into the y equals and you and we can work with it. So what we want to do is use this equation to estimate the population in 1925 and predict the population in the year 2020 which just so happens to be the year in which I'm talking right now. So what we can do is use the table for that. So I'm going to go into table set and make sure that my independent variable is set to ask, and that will allow me to type in my own X values. Once that is set, I can go into the table, which is second graf and it doesn't matter if you have numbers here. You can delete them if you want, but it's not going to matter. So for the year 1925 we want to type in a 25 x, And that tells us 110.82 is the population of the United States in millions, according to this model from the year 1925. And now let's type in the year 1 20 to represent the year 2020 and we get 367 million for the population in the United States.