SOCIAL SCIENCE: Immigration The percentage of immigrants in the United States has changed...

SOCIAL SCIENCE: Immigration The percentage of immigrants in the United States has changed since World War I as shown in the following table.

a. Number the data columns with x-values 0–7 (so that x stands for decades since 1930) and use quadratic regression to fit a parabola to the data. State the regression function. [Hint: See Example 10.]

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b. In 2007, one in eight people living in the United States was an immigrant. Use the regression function to predict the percentage in 2007, How well does this prediction match the actual percentage?

Sources: Center for Immigration Studies and U.S. Census Bureau

Example 10 QUADRATIC REGRESSION USING A GRAPHING CALCULATOR

The following graph gives the total annual sales of Macintosh computers from 2001 through 2007.

Source: systemshootouts.com

a. Use quadratic regression to fit a parabola to the data and state the regression function.

b. Use the regression function to predict Macintosh sales in 2012.

Solution

a. We number the years with x-values 0–7, so x stands for years after 2000. We enter the data into lists, as shown in the first screen below (as explained in Graphing Calculator Basics—Entering Data on page xxiii), and use ZoomStat to graph the data points.

Then (using STAT, CALC, and QuadReg) we graph the regression along with the data points.

The regression curve, which fits the points reasonably well, is

y = 0.20x2 – 0.88x + 3.9 Rounded

b. To predict sales in 2012, we evaluate Y1 at 12 (since x = 12 corresponds to 2012). From the screen on the left, if the current trend continues, Macintosh sales in 2012 will be approximately 22.8 million units.