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Unit Problem Write-Up 04.08-09.13 (10 points in the 4th quarter product column)

In this guided worksheet you will be solving the unit problem. Recall that the unit problem asked us to use a mathematical model to determine when the population on the planet will exceed 1.6 X 1015. This is roughly one person per square foot of land (regardless of terrain).

I)Using technology, make a scatterplot of the data. (We did this the first day of the unit).

Years since 1650 / Population
In millions
0.1 / 470
100 / 694
200 / 1091
250 / 1570
300 / 2510
310 / 3030
320 / 3680
330 / 4480
335 / 4870
340 / 5290
345 / 5730
360 / 7000

II)“Sketch” the scatter plot on graph paper. Label and scale both the x and y axis.

III)In the statistical world, predictions are made using mathematical models that best fit the sample data. The data we have is called bivariate because there are two variables x and y. We want to create an equation that we can use to predict the population by doing math to the number of years since 1650. This equation is called a Regression.

The above data is called a sample because it does not include every population count for every year, day, second etc. In fact, in statistics, things like global population are considered to be continuous measurements since the population is so large and constantly fluxuates .

We will begin by attempting to straighten our curved data.

IV)Recall from our earlier work that our data looks exponential. An exponential curve can often be straightened by taking the log or lnof the y data.

Use List 3 (L3) to transform the data by taking ln(L2).

Make a scatter plot of (L1, L3) in your calculator and draw a rough sketch. Does the scatter plot appear straighter?

V)We want the data to be as straight as possible. Now the data have a slight bend. Experiment with the data in L3 by transforming the values and storing them in L5. Follow class instructions on how to do this repeatedly. Draw a rough sketch for each transformation. Think about why each transformation is or is not a reasonable experiment.

Sketch

VI)Continue to experiment with various transformations until you think the data is as straight as you can get it. When you have finished, write down the transformation below as a mathematical expression similar to what you see in part V.

VII)We will now fit an equation (regression) to the straightened values. Follow class instructions on how to do this. Write the result of your equation below.

VIII)Use your equation to find the year when the population will hit the target of the unit problem. SHOW ALL WORK BELOW.