Pre-Calc Hour ____ Name _____________________

Internet Activity: How Fast is the Population Growing?

 In this activity, you will access population data from the US Census Bureau Internet sight. You will analyze the data to help you predict the rate in which the US and Wisconsin populations are growing and also to predict their populations in the past and in the future.

Part A: Using the Internet

You can use the Internet to help you find the necessary information to complete this task. Open the Internet program Netscape and enter the following address: www.census.gov/population/www/censusdata/cencounts.html

Part B: Collecting Data

Once you have found the Census Bureau on the Internet, select the state of Wisconsin, and complete the data tables below. In the year column, please note that 0 = 1900, 10 = 1910, 20 = 1920 and so on.

Year (L1)	Population of the US (L2)	Population of Wisconsin (L3)
0
10
20
30
40
50
60
70
80
90

 After you have copied all of the data from the Internet onto the worksheet, input the data into your calculator. Assign year to L₁, population of the United States to L₂, and population of Wisconsin to L₃.

Part C: Analysis of US data

• Set up the STAT PLOT menu on your calculator to graph the Population of the United States vs. the Year.

• Now find an exponential model to describe the data and store it directly to Y₁ in the Y= menu on your calculator. Graph the data and exponential model to make sure that the model passes near the cluster of data points. Answer the following questions about the population of the United States based on your model.

1. By what percent does the population of the United States increase each year? Explain how you found this answer.

2. State the initial value "a" in this situation and explain what it represents. Does this seem reasonable? Explain why or why not.

3. Predict the population of the United States for 1999. Show work.

4. Predict the population of the United States for 1850. Show work.

5. In what year will the population of the United States reach 300 million? Show work.

Part D: Does population grow exponentially?

• Perform a linear regression of the US population data vs. the Year and store the equation in Y₂.
• Perform a quadratic regression of the US population data vs. the Year and store the equation in Y₃.
• Perform a cubic regression of the US population data vs. the Year and store the equation in Y₄.

To find which equation best represents the US population from 1900 to 1990, we will have to find the sum of the squares of the errors (deviations) of the data points from each function. This operation can be performed on your calculator. First you have to find the errors between the observed data and predicted data (based on your equations). Press STAT and ENTER. Go to the top of L₄. Press 2^nd, L2, -, 2^nd, VARS, choose "1: Function," choose "1:Y1,: left parenthesis, 2^nd, L1, right parenthesis. It should look like: L₂ – Y₁(L₁), press ENTER. These values represent the error of the exponential model stored in Y₁. Do the same thing for the linear, quadratic, and cubic regressions. Store the errors in the remaining lists, create a new list if you have to.

 Go to the home screen and press STAT, arrow over to "CALC," choose "1:1 VAR STATS" and press ENTER. Press 2^nd 4 to get L₄ to appear on the screen. Press ENTER. Scroll down to where is says S x². That is the sum of the square of the errors. Record each of these values in the table below.

model (type)	equation (all decimals)	sum of squared errors
exponential
linear
quadratic
cubic

6. Which model best describes the data and how do you know (interpolation)? Which model is best for making predictions about population in the past or future (extrapolation)? Explain your reasoning.

Part E: Homework

Repeat questions 1-6 with the Wisconsin state population data, and determine the year when the population of the state will reach 15 million (question 5). Be sure to select columns L₁ to describe the x-values and L₃ to describe the y-values in your calculations. All of your work should be on a separate sheet of loose-leaf paper. Be sure to include a table of the sum of the square of the errors for all the models.