To find the linear regression equation of the form $y = mx + b$ using the given data, we will calculate the slope ($m$) and y-intercept ($b$) using the least squares method. You can also use a calculator, spreadsheet, or statistical software to perform this. Here's the manual step-by-step method:

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Given data:

$x$	$y$
1	4.48
2	5.89
3	5.14
4	7.35
5	7.39
6	6.4
7	8.8
8	8.53
9	12.65

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Step 1: Formulas for slope ($m$) and intercept ($b$):

$m = \frac{n \sum (x y) - \sum x \sum y}{n \sum (x^2) - (\sum x)^2}$ $b = \frac{\sum y - m \sum x}{n}$ Where $n$ is the number of data points, $\sum x$, $\sum y$, $\sum (xy)$, and $\sum (x^2)$ are summations over the data.

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Step 2: Compute necessary sums:

We calculate: $\sum x, \quad \sum y, \quad \sum (x^2), \quad \sum (xy)$

$x$	$y$	$x^2$	$x \cdot y$
1	4.48	1	4.48
2	5.89	4	11.78
3	5.14	9	15.42
4	7.35	16	29.4
5	7.39	25	36.95
6	6.4	36	38.4
7	8.8	49	61.6
8	8.53	64	68.24
9	12.65	81	113.85

Summations:

$\sum x = 45, \quad \sum y = 66.63, \quad \sum (x^2) = 285, \quad \sum (xy) = 379.12$

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Step 3: Calculate slope ($m$):

Using the formula: $m = \frac{n \sum (xy) - \sum x \sum y}{n \sum (x^2) - (\sum x)^2}$

Substitute: $n = 9, \quad \sum (xy) = 379.12, \quad \sum x = 45, \quad \sum y = 66.63, \quad \sum (x^2) = 285$

$m = \frac{9(379.12) - (45)(66.63)}{9(285) - (45)^2}$ $m = \frac{3412.08 - 2998.35}{2565 - 2025}$ $m = \frac{413.73}{540} \approx 0.7662$

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Step 4: Calculate intercept ($b$):

Using the formula: $b = \frac{\sum y - m \sum x}{n}$ Substitute: $b = \frac{66.63 - (0.7662)(45)}{9}$ $b = \frac{66.63 - 34.479}{9}$ $b = \frac{32.151}{9} \approx 3.57$

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Final Answer:

The equation of the linear regression line is: $y = 0.77x + 3.57$

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Would you like further explanation or additional details on this process?

Follow-up questions:

1. What is the significance of the slope $m$ in a linear regression model?
2. How can you interpret the y-intercept $b$ in this context?
3. Would a scatter plot of this data show a strong linear relationship? Why?
4. How do residuals help determine the quality of a regression model?
5. Can you explain how rounding affects the precision of predictions using the regression line?

Tip:

Using a graphing calculator, spreadsheet software (like Excel or Google Sheets), or statistical tools (like Python or R) can simplify linear regression calculations significantly!