Questions: In examining per capita incomes for U.S. states over time, one often finds that "poor" states tend to stay poor and "wealthy" states tend to stay wealthy. So, predictions about later years may sometimes be made from knowledge of previous years. Consider the 1999 per capita income for a state (denoted by y) and its 1980 per capita income (denoted by x). The following bivariate data give the per capita income (in thousands of dollars) for a sample of sixteen states in the years 1980 and 1999 (source: U.S. Bureau of Economic Analysis, Survey of Current Business, May 2000). The data are plotted in the scatter plot in Figure 1. - 1980 per capita income, x (in 1000s) - 1999 per capita income, y (in 1000s) Alabama 7.9 22.9 Iowa 9.7 25.7 Rhode Island 9.7 29.7 South Dakota 8.1 25.1 Montana 9.1 22.3 Michigan 10.4 27.8 Nebraska 9.3 27.4 Washington 10.9 30.3 Missouri 9.4 26.2 Utah 8.5 23.4 New Mexico 8.4 22.1 Tennessee 8.3 25.6 Maryland 11.2 32.2 Pennsylvania 10.2 28.7 Virginia 10.2 29.5 Oregon 10.2 27.1 1980 per capita income (in 1000s) Figure 1 The value of the sample correlation coefficient r for these data is approximately 0.846.

In examining per capita incomes for U.S. states over time, one often finds that "poor" states tend to stay poor and "wealthy" states tend to stay wealthy. So, predictions about later years may sometimes be made from knowledge of previous years. Consider the 1999 per capita income for a state (denoted by y) and its 1980 per capita income (denoted by x). The following bivariate data give the per capita income (in thousands of dollars) for a sample of sixteen states in the years 1980 and 1999 (source: U.S. Bureau of Economic Analysis, Survey of Current Business, May 2000). The data are plotted in the scatter plot in Figure 1. - 1980 per capita income, x (in 1000s) - 1999 per capita income, y (in 1000s) Alabama 7.9 22.9 Iowa 9.7 25.7 Rhode Island 9.7 29.7 South Dakota 8.1 25.1 Montana 9.1 22.3 Michigan 10.4 27.8 Nebraska 9.3 27.4 Washington 10.9 30.3 Missouri 9.4 26.2 Utah 8.5 23.4 New Mexico 8.4 22.1 Tennessee 8.3 25.6 Maryland 11.2 32.2 Pennsylvania 10.2 28.7 Virginia 10.2 29.5 Oregon 10.2 27.1 1980 per capita income (in 1000s) Figure 1 The value of the sample correlation coefficient r for these data is approximately 0.846.
Transcript text: Correlation and Simple Linear Regression Relating the sample correlation coefficient and the parameters of the lea... QUESTION In examining per capita incomes for U.S. states over time, one often finds that "poor" states tend to stay poor and "wealthy" states tend to stay wealthy. So, predictions about later years may sometimes be made from knowledge of previous years. Consider the 1999 per capita income for a state (denoted by $y$ ) and its 1980 per capita income (denoted by $x$ ). The following bivariate data give the per capita income (in thousands of dollars) for a sample of sixteen states in the years 1980 and 1999 (source: U.S. Bureau of Economic Analysis, Survey of Current Business, May 2000). The data are plotted in the scatter plot in Figure 1. \begin{tabular}{|c|c|c|} \hline & 1980 per capita income, $x$ (in \$1000s) & 1999 per capita income, $y$ (in \$1000s) \\ \hline Alabama & 7.9 & 22.9 \\ \hline Iowa & 9.7 & 25.7 \\ \hline Rhode Island & 9.7 & 29.7 \\ \hline South Dakota & 8.1 & 25.1 \\ \hline Montana & 9.1 & 22.3 \\ \hline Michigan & 10.4 & 27.8 \\ \hline Nebraska & 9.3 & 27.4 \\ \hline Washington & 10.9 & 30.3 \\ \hline Missouri & 9.4 & 26.2 \\ \hline Utah & 8.5 & 23.4 \\ \hline New Mexico & 8.4 & 22.1 \\ \hline Tennessee & 8.3 & 25.6 \\ \hline Maryland & 11.2 & 32.2 \\ \hline Pennsylvania & 10.2 & 28.7 \\ \hline Virginia & 10.2 & 29.5 \\ \hline Oregon & 10.2 & 27.1 \\ \hline \end{tabular} 1980 per capita income (in $\$ 1000$ s) Figure 1 The value of the sample correlation coefficient $r$ for these data is approximately 0.846 .
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Solution

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Solution Steps

Step 1: Understanding the Problem

The problem involves examining the relationship between the 1980 per capita income (x) and the 1999 per capita income (y) for a sample of sixteen U.S. states. The goal is to use the given bivariate data to make predictions about future per capita incomes based on past data.

Step 2: Identifying the Given Data

The given data includes:

  • 1980 per capita income (x) in thousands of dollars
  • 1999 per capita income (y) in thousands of dollars

The data for the sixteen states is as follows:

| State | 1980 per capita income, x (in $1000s) | 1999 per capita income, y (in $1000s) | |---------------|---------------------------------------|---------------------------------------| | Alabama | 7.9 | 22.9 | | Iowa | 9.7 | 25.7 | | Rhode Island | 9.7 | 29.7 | | South Dakota | 8.1 | 25.1 | | Montana | 9.1 | 22.3 | | Michigan | 10.4 | 27.8 | | Nebraska | 9.3 | 27.4 | | Washington | 10.9 | 30.3 | | Missouri | 9.4 | 26.2 | | Utah | 8.5 | 23.4 | | New Mexico | 8.4 | 22.6 | | Tennessee | 8.3 | 25.6 | | Maryland | 11.2 | 32.6 | | Pennsylvania | 10.2 | 29.2 | | Virginia | 10.2 | 29.5 | | Oregon | 10.2 | 27.1 |

Step 3: Calculating the Sample Correlation Coefficient

The sample correlation coefficient (r) measures the strength and direction of the linear relationship between the two variables. The value of r for these data is approximately 0.846, indicating a strong positive linear relationship.

Final Answer

The sample correlation coefficient (r) for the given data is approximately 0.846. This indicates a strong positive linear relationship between the 1980 per capita income and the 1999 per capita income for the sample of sixteen U.S. states.

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