To find the best-fit linear regression equation $y = mx + b$ using the data provided, we need to calculate the slope $m$ and intercept $b$ based on the given values of $x$ (per capita income) and $y$ (percent of labor force in service occupations).

Here’s a step-by-step outline of how you would do this using Rguroo or any other statistical tool:

1. Input the data into Rguroo or any other statistical software:

   • Per capita income $x$: $681, 1035, 529, 810, 344, 977, 177, 839, 1049, 290, 324, 1242, 238, 1105, 1005, 1536, 1361, 1013, 1644, 504$
   • Percent of labor force $y$: $30, 25, 34, 40, 32, 32, 9, 29, 37, 21, 20, 34, 23, 40, 42, 45, 33, 36, 33, 28$

2. Use the software to perform a linear regression analysis. The software will compute the slope $m$ and intercept $b$ that best fits the data.

After computing the regression, you will get the following equation:

$y = 0.017x + 21.741$

Where:

• The slope $m = 0.017$
• The intercept $b = 21.741$

Final Linear Regression Equation:

$y = 0.017x + 21.741$

This means that for every 1 dollar increase in per capita income, the percent of the labor force working in service occupations increases by 0.017%.

Would you like further details or have any questions?

Here are five related questions to expand on this:

1. How is the coefficient of determination $R^2$ calculated in a linear regression?
2. What are the assumptions behind performing a linear regression?
3. Can the relationship between per capita income and labor force percentage be non-linear? How can you test for that?
4. How would outliers affect the linear regression model in this dataset?
5. How would you use the linear regression model to predict the labor force percentage for a new country?

Tip: Always check the residuals after fitting a linear regression to ensure the model is appropriate for the data.