Given below is the per capita income, x, in dollars for certain countries and the percent of the labor force that works in service occupations, y, in a certain year. Use Rguroo to find the best fit linear regression equation. Round the slope and intercept to three decimal places.

x	y

681	30

1035	25

529	34

810	40

344	32

977	32

177	9

839	29

1049	37

290	21

324	20

1242	34

238	23

1105	40

1005	42

1536	45

1361	33

1013	36

1644	33

504	28

Provide your answer below:

y =  x +

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%.

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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.

Given below is the per capita income, x, in dollars for certain countries and the percent of the labor force that works in service occupations, y, in a certain year. Use Rguroo to find the best fit linear regression equation. Round the slope and intercept to three decimal places.

Learn how to find the best fit linear regression equation for per capita income and labor force percentage in service occupations. This problem involves using the linear regression formula y = mx + b, suitable for high school students in Grades 10-12.

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