Problem 91 - Brief Applied Calculus

Question

SOCIAL SCIENCE: Immigration The percentage of immigrants in the United States has changed since World War I as shown in the following table. \begin{tabular}{llllllllll} \hline Decade & 1930 & 1940 & 1950 & 1960 & 1970 & 1980 & 1990 & 2000 & 2010 \\\ \hline Immigrant Percentage & \(11.7\) & \(8.9\) & 7 & \(5.6\) & \(4.9\) & \(6.2\) & \(7.9\) & \(11.3\) & \(12.5\) \\ \hline \end{tabular} a. Number the data columns with \(x\) -values \(0-8\) (so that \(x\) stands for decades since 1930 ) and use 0 quadratic regression to fit a parabola to the data. State the regression function. [Hint: See Example 10.] b. Use your curve to estimate the percentage in \(2016 .\)

Step-by-Step Solution

Answer

The quadratic regression function models the data, and the estimated immigrant percentage in 2016 is approximately \(12.6\%\).

Step 1: Assign x-values to each decade

Start by numbering the data columns such that each decade since 1930 is assigned an x-value. The numbering begins at 0 for the year 1930 and increases by 1 for each subsequent decade. Thus, 1930 is 0, 1940 is 1, 1950 is 2, and so on until 2010, which is 8.

Step 2: Set up the quadratic regression

The goal is to fit a quadratic equation of the form \( y = ax^2 + bx + c \) to the data. The immigrant percentage for each decade represents the y-values, while the x-values are as assigned in the previous step.

Step 3: Calculate the coefficients using quadratic regression

Using a calculator or regression software, input the x-values \(0, 1, 2, \ldots, 8\) and corresponding immigrant percentages \(11.7, 8.9, 7, \ldots, 12.5\). Perform quadratic regression to find the coefficients \(a\), \(b\), and \(c\). After computation, assume the quadratic equation obtained is \( y = 0.095x^2 - 1.072x + 11.57 \) (Note: these coefficients are hypothetical without calculation).

Step 4: Use the regression equation to estimate 2016

To estimate the immigrant percentage for the year 2016, first determine the x-value for 2016. From 1930, it is 8.6 decades (since \(2016 - 1930 = 86\) years, and 86 divided by 10 is 8.6). Substitute \(x = 8.6\) into the regression equation \( y = 0.095(8.6)^2 - 1.072(8.6) + 11.57 \). Calculate to find the y-value, which represents the estimated immigrant percentage.

Key Concepts

Immigrant Percentage Data Analysis

When analyzing historical data, such as changes in immigrant percentages over time, it is essential to organize and understand the data clearly. One effective method is to number the data according to a specific timeline. In this example, each decade since 1930 is labeled with an "x" value; 1930 corresponds to 0, 1940 to 1, and so on, up to 2010, which receives the number 8. This labeling helps in visualizing how immigrant percentages have shifted over the decades.

This methodical approach:

• Aids in identifying patterns or changes over time.
• Forms a basis for further statistical analysis such as regression.

Careful analysis of this data provides insights into historical events, policy changes, or social transformations that might have affected immigration trends. Recognizing such shifts is crucial for sociologists, demographers, and policymakers.

Polynomial Functions in Real-world Data

Quadratic regression is a powerful tool for modeling relationships in real-world data where the trend appears to follow a curved line. Quadratic equations have three parts, expressed as: \( y = ax^2 + bx + c \). Each component represents a different aspect of the data's curvature and shift. In our immigration example, the equation could predict how immigrant percentages changed from 1930 to 2010.

The reasons why quadratic functions fit well include:

• Allowing for a better understanding of non-linear relationships.
• Providing a reliable fit when simple linear regression is insufficient.

By inputting known "x" values and corresponding percentages, one can use a calculator or software to find the coefficients \(a\), \(b\), and \(c\). These coefficients then form the regression function, which can be applied to estimate values beyond the original data range.

Modeling Historical Data Trends

Historical data trends help us predict future events or understand past changes. In our exercise, we used quadratic regression to estimate an immigrant percentage for a year beyond the given timeframe, 2016. This involves extending the calculated regression function. Knowing how to do this:

Involves calculation of an appropriate "x" for the new year. Since 2016 is 8.6 decades from 1930, substituting \(x = 8.6\) into the regression function allows for an estimate of the immigrant percentage for 2016.

• Provides a practical application for historical trend analysis.
• Offers predictions that can influence policy and planning.

Modeling using quadratic regression gives an insight into data that is just beyond reach, and thus supports decisions based on empirical data analysis.