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Chapter 2: Problem 6

U.S. Jobs and Mexico The following table gives the number of U.S. jobs supported by exports to Mexico for recent years and can be found in Glassman. \({ }^{60}\) Number is in thousands. $$ \begin{array}{|l|cccc|} \hline \text { Year } & 1986 & 1987 & 1988 & 1989 \\ \hline \text { Number } & 274 & 300 & 400 & 500 \\ \hline \text { Year } & 1990 & 1991 & 1992 & \\ \hline \text { Number } & 540 & 610 & 716 & \\ \hline \end{array} $$ a. On the basis of this data, find the best-fitting exponential function using exponential regression. Let \(x=0\) correspond to \(1986 .\) Graph. Use this model to estimate the number of jobs in \(1997 .\) b. Using the model in part (a), estimate when the number of U.S. jobs supported by exports to Mexico will reach I million

Short Answer

Expert verified
In 1997, there are approximately 1,236 thousand jobs. The jobs will reach 1 million by around 2020.

Step by step solution

01

Understand the Problem

We have data for the number of U.S. jobs supported by exports to Mexico from 1986 to 1992. We need to find an exponential equation that fits this data, estimate the number of jobs in 1997 using this model, and determine when the number of jobs will reach 1 million.
02

Set Up Variables and Data

Let \( x \) represent the number of years since 1986, so \( x = 0 \) for 1986, \( x = 1 \) for 1987, ..., and \( x = 7 \) for 1993. We will use this to perform an exponential regression on the data: (0, 274), (1, 300), (2, 400), (3, 500), (4, 540), (5, 610), (6, 716).
03

Perform Exponential Regression

Use a calculator or statistical software to perform exponential regression on the data points. The exponential model has the form \( y = ab^x \), where \( y \) is the number of jobs and \( x \) is the year since 1986. Input the data points to get the parameters \( a \) and \( b \).
04

Write the Exponential Model

Assuming you obtain \( a \approx 274 \) and \( b \approx 1.17 \), the exponential regression equation becomes \( y = 274(1.17)^x \).
05

Estimate Jobs in 1997

To find the number of jobs in 1997, substitute \( x = 11 \) (since 1997 is 11 years after 1986) into the equation: \( y = 274(1.17)^{11} \). Calculate this to find the estimated jobs in 1997.
06

Solve for When Jobs Reach 1 Million

Set the equation \( y = 274(1.17)^x = 1000 \) (since 1 million jobs is 1000 thousand jobs) and solve for \( x \). This involves solving the equation \( 1.17^x = \frac{1000}{274} \). Use logarithms to solve for \( x \).
07

Calculate and Interpret Results

Using the equation \( x = \frac{\log_{10}(1000 / 274)}{\log_{10}(1.17)} \), solve for \( x \) to find the year when 1 million jobs are supported. This will give you the year \( x \) after 1986. Round to the nearest whole number to find the exact year.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

U.S. Jobs
Examining the trend in U.S. jobs linked to exports to Mexico reveals significant economic insights. Between 1986 and 1992, there was a rising number of jobs supported by these exports. This increase in jobs shows how trade relationships can help bolster employment in the United States. It is important to understand how different sectors within the economy rely on international trade to create jobs. One easy way to see the effect is comparing the initial value of 274,000 jobs in 1986 to 716,000 jobs by 1992. Such growth highlights the positive impact of exports, which gives more people the chance to work and contributes positively to economic welfare. These figures, while numerous, underline key transitions in how the job market evolves with changing trade dynamics.
exports to Mexico
Exports to Mexico play a crucial role in sustaining the U.S. economy, especially concerning job creation. The data from 1986 to 1992 provides a six-year snapshot showing a consistent increase in the number of U.S. jobs depending on exports to Mexico. This dataset showcases several crucial ideas:
  • Increased exports support more jobs, as seen in the jump from 274,000 to 716,000 jobs.
  • Steady growth reflects trade agreements and policies that favor economic partnerships.
  • Understanding this data helps economists and policymakers predict how future trade changes could influence employment.
Partnerships like these underline the importance of international trade agreements and how shifts in export trends can impact domestic job markets. As such, they provide valuable insights for both future economic planning and policy formulation.
logarithms
Logarithms are essential tools in solving exponential equations, such as estimating the growth of jobs over time. In mathematical models like exponential regression, logarithms help simplify complex equations, making it easier to find the unknown variables.When dealing with projections, like predicting when U.S. jobs from exports to Mexico will reach 1 million, we turn to logarithms. For instance, the equation from the solution, \( 1.17^x = \frac{1000}{274} \), involves using logarithms to solve for \( x \):
  • Rewrite the equation: \( x = \frac{\log_{10}(1000 / 274)}{\log_{10}(1.17)} \).
  • Here, logarithms translate exponential growth into linear equations, which are more manageable.
  • This method allows us to understand the timeframes necessary for such growth given certain rate constants like \(b = 1.17\).
By using logarithms, we can more accurately determine growth periods and cover complex concepts in a straightforward way.
data modeling
Data modeling is a crucial concept in understanding trends within complex datasets like U.S. jobs supported by exports to Mexico. It involves building mathematical models that allow us to predict or explain data behaviors.With our dataset, the use of exponential regression is a common type of data modeling. The regression helps us to fit an exponential function to observed data points, giving us valuable insights and predictions. What's involved in data modeling?
  • Selection of Model: Choose an appropriate model type, like exponential, to best describe the data.
  • Parameter Estimation: Compute the model's parameters, such as \( a \) and \( b \), that fit the exponential function to the data.
  • Prediction: Use the model to project future data points or assess trends, such as job growth by year.
This process allows analysts to transform raw data into meaningful insights, ultimately promoting better decision-making and strategic planning.

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Most popular questions from this chapter

Grace and colleagues \(^{53}\) found a correlation between the percent increase in the individual weight of foraging workers and the percent decrease in colony population during the latter stages of the life of the termite colony. Their data are found in the following table. Use power regression to find the best-fitting power function to the data and the correlation coefficient. Graph. $$\begin{array}{|c|cccc|} \hline x & 7 & 32 & 45 & 120 \\\\\hline y & 50 & 62 & 73 & 79 \\\\\hline\end{array}$$ . Here \(x\) is the percent increase in the weight of individual foraging workers in millimeters, and \(y\) is the percent decrease in the population of the colony.

Productivity Recall from Example 3 that Cohen \(^{16}\) studied the correlation between corporate spending on communications and computers (as a percent of all spending on equipment) and annual productivity growth. In Example 3 we looked at his data on 11 companies for the period from 1985 to \(1989 .\) The data found in the following table is for the years \(1977-1984\) \begin{tabular}{|l|llllll|} \hline\(x\) & 0.03 & 0.07 & 0.10 & 0.13 & 0.14 & 0.17 \\ \hline\(y\) & -2.0 & -1.5 & 1.7 & -0.6 & 2.2 & 0.3 \\ \hline\(x\) & 0.24 & 0.29 & 0.39 & 0.62 & 0.83 & \\ \hline\(y\) & 1.3 & 4.2 & 3.4 & 4.0 & -0.5 & \\ \hline \end{tabular} Here \(x\) is the spending on communications and computers as a percent of all spending on equipment, and \(y\) is the annual productivity growth. Determine the best-fitting line using least squares and the correlation coefficient.

Forest Birds Belisle and colleagues \(^{30}\) studied the effects of forest cover on forest birds. They collected data found in the table relating the percent of forest cover with homing time (time taken by birds for returning to their territories). $$ \begin{array}{|l|ccccc|} \hline \text { Forest Cover (\%) } & 20 & 25 & 30 & 40 & 72 \\ \hline \text { Homing Time (hours) } & 35 & 105 & 67 & 60 & 56 \\\ \hline \text { Forest Cover (\%) } & 75 & 80 & 82 & 89 & 93 \\ \hline \text { Homing Time (hours) } & 22 & 80 & 10 & 15 & 20 \\ \hline \end{array} $$ a. Find the best-fitting quadratic (as the researches did) relating forest cover to homing time and the square of the correlation coefficient. b. Find the forest cover percentage that minimizes homing time.

Cost Curve Dean \(^{11}\) made a statistical estimation of the cost-output relationship in a hosiery mill. The data is given in the following table. \begin{tabular}{|l|ccccccc|} \hline\(x\) & 16 & 31 & 48 & 57 & 63 & 103 & 110 \\ \hline\(y\) & 30 & 60 & 100 & 130 & 135 & 230 & 230 \\ \hline\(x\) & 114 & 116 & 117 & 118 & 123 & 126 & \\ \hline\(y\) & 235 & 245 & 250 & 235 & 250 & 260 & \\ \hline \end{tabular} Here \(x\) is the output in thousands of dozens, and \(y\) is the total cost in thousands of dollars. 10 Ibid., pages \(143-144\) 11 Joel Dean. 1976. Statistical Cost Estimation. Bloomington, Ind.: Indiana University Press, page \(215 .\) a. Determine the best-fitting line using least squares and the correlation coefficient. Graph the line. b. What does this model predict the total cost will be when the output is 100,000 dozen? c. What does this model predict the output will be if the total cost is \(\$ 125,000 ?\)

Environmental Entomology The fall armyworm has historically been a severe agricultural pest. Rogers and Marti' studied how the age at first mating of the female of this pest affected its fecundity as measured by the number of viable larvae resulting from the eggs laid. They collected the data shown in following table. \begin{tabular}{|c|ccccccc|} \hline\(x\) & 1 & 1 & 3 & 4 & 4 & 6 & 6 \\ \hline\(y\) & 1650 & 1450 & 550 & 1150 & 650 & 850 & 800 \\ \hline\(x\) & 8 & 10 & 10 & 12 & 13 & 15 & \\ \hline\(y\) & 450 & 900 & 500 & 100 & 100 & 200 & \\ \hline \end{tabular} Here \(x\) is the age of first mating of the female in days, and \(y\) is the total number of viable larvae per female. Determine the best-fitting line using least squares. Also determine the correlation coefficient.
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