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The problem asks to fit a simple linear regression equation \( y = a + bx \) to the given data using the method of least squares. The table provides the number of man-hours (\( x \)) and corresponding productivity (\( y \)):

| Man-hours (\( x \)) | Productivity (\( y \)) |
|---------------------|------------------------|
| 3.6                 | 9.5                    |
| 4.8                 | 10.2                   |
| 6.2                 | 11.5                   |
| 6.9                 | 12.2                   |
| 10.1                | 16.8                   |
| 6.1                 | 13.2                   |
| 7.9                 | 20.5                   |
| 9.5                 | 22.7                   |
| 5.4                 | 12.7                   |

To apply the least squares method, we will calculate:

1. The means \( \bar{x} \) and \( \bar{y} \),
2. The slope \( b \) using the formula:
   \[
   b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
   \]
3. The intercept \( a \) using:
   \[
   a = \bar{y} - b\bar{x}
   \]

Let me calculate this now.The linear regression equation that fits the data using the method of least squares is:

\[
y = 2.207 + 1.809x
\]

This means that for each additional man-hour, the productivity increases by approximately 1.809 units.

Do you have any questions or would you like further details?

Here are five related questions for further exploration:
1. How would you interpret the slope in this context?
2. What is the coefficient of determination \( R^2 \) for this regression line?
3. How would you use this regression equation to predict productivity for 8 man-hours?
4. Can we apply this method to non-linear data? Why or why not?
5. How would you verify the accuracy of this regression model?

**Tip**: Always check if the residuals (differences between actual and predicted values) are randomly scattered to ensure that a linear model is appropriate.

Fit a simple linear regression equation y = a + bx by applying the method of least squares based on the provided data of man-hours and corresponding productivity.

Learn how to fit a simple linear regression equation to the given data of man-hours and productivity using the least squares method. This solution covers important concepts such as linear regression, the slope formula, and intercept calculation. Suitable for high school and introductory college-level statistics.

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