To find the intercept of the line of best fit using the least squares regression method, we'll use the following approach:

Step 1: List the data points

The data pairs $(x, y)$ are:

$(2, 6), (4, 9), (6, 15), (10, 18), (12, 11), (14, 19), (15, 24), (16, 21), (20, 29)$

Step 2: Apply the Least Squares Regression Formula

The equation of the line of best fit is:

$y = mx + b$

Where:

• $m$ is the slope of the line.
• $b$ is the intercept of the line.

We calculate the slope $m$ using the formula:

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

Then, to find the intercept $b$, we use the formula:

$b = \frac{\sum y - m (\sum x)}{n}$

Where:

• $n$ is the number of data points.
• $\sum x$, $\sum y$, $\sum xy$, and $\sum x^2$ are the sums of $x$, $y$, the products $xy$, and the squares of $x$, respectively.

Let's compute these values step-by-step.

Step 3: Compute the necessary sums

$\sum x = 2 + 4 + 6 + 10 + 12 + 14 + 15 + 16 + 20 = 99$ $\sum y = 6 + 9 + 15 + 18 + 11 + 19 + 24 + 21 + 29 = 152$ $\sum xy = (2 \times 6) + (4 \times 9) + (6 \times 15) + (10 \times 18) + (12 \times 11) + (14 \times 19) + (15 \times 24) + (16 \times 21) + (20 \times 29) = 1940$ $\sum x^2 = 2^2 + 4^2 + 6^2 + 10^2 + 12^2 + 14^2 + 15^2 + 16^2 + 20^2 = 1298$ $n = 9 \quad (\text{number of data points})$

Step 4: Calculate the slope $m$

$m = \frac{9(1940) - (99)(152)}{9(1298) - (99)^2}$ $m = \frac{17460 - 15048}{11682 - 9801}$ $m = \frac{2412}{1881} \approx 1.28$

Step 5: Calculate the intercept $b$

$b = \frac{\sum y - m (\sum x)}{n}$ $b = \frac{152 - 1.28 \times 99}{9}$ $b = \frac{152 - 126.72}{9}$ $b = \frac{25.28}{9} \approx 2.81$

Final Answer:

The intercept of the line of best fit is approximately 2.81.

Would you like more details or have any questions?

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Here are five related questions:

1. How would you calculate the slope using different sets of points?
2. What is the significance of the intercept in the context of linear regression?
3. How can outliers affect the line of best fit?
4. How can you graphically verify the line of best fit?
5. How do you compute the correlation coefficient for this data?

Tip: Always check the residuals (the differences between the observed and predicted values) to ensure the line of best fit appropriately models the data.