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To compute the least-squares regression equation for the given data set, we need to find the slope (\(m\)) and the y-intercept (\(b\)) using the formula for the line of best fit, \( \hat{y} = mx + b \). 

The data provided are:
\[
x: 5.8, 4.1, 6.5, 4.6, 6.6, 5.8, 5.2
\]
\[
y: 2.0, 4.9, 1.0, 4.1, 1.2, 1.9, 2.9
\]

### Steps to calculate the least-squares regression equation:

1. **Calculate the means** of \( x \) and \( y \):
   \[
   \bar{x} = \frac{\sum x}{n}, \quad \bar{y} = \frac{\sum y}{n}
   \]
   
2. **Calculate the slope** \( m \) using:
   \[
   m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
   \]

3. **Calculate the y-intercept** \( b \) using:
   \[
   b = \bar{y} - m\bar{x}
   \]

Let me work on this and provide you with the regression equation.The least-squares regression equation for the given data set is:

\[
\hat{y} = -1.5599x + 11.1731
\]

This equation indicates that the slope is approximately \(-1.5599\), and the y-intercept is approximately \(11.1731\).

Would you like further details or have any questions? Here are five related questions you might consider:

1. What does the negative slope of \(-1.5599\) indicate about the relationship between \(x\) and \(y\)?
2. How can we calculate the correlation coefficient to measure the strength of this relationship?
3. Can we use this equation to make predictions for new values of \(x\)?
4. What is the importance of rounding the slope and intercept to four decimal places in regression analysis?
5. How would outliers in the data affect the regression equation?

**Tip:** When interpreting regression results, the slope tells you the change in the dependent variable \(y\) for each unit increase in the independent variable \(x\).

Compute the least-squares regression equation for the given data set. Round the slope and y-intercept to at least four decimal places. The data is as follows:
x: [5.8, 4.1, 6.5, 4.6, 6.6, 5.8, 5.2]
y: [2.0, 4.9, 1.0, 4.1, 1.2, 1.9, 2.9]

Learn how to compute the least-squares regression equation for a given data set. This example includes data points for x and y, and shows how to calculate the slope and y-intercept using the least-squares method. Suitable for advanced high school or college-level students.

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