To find the regression line for the given data, we need to use the formula for the least squares regression line:

$y = mx + b$

Where:

• $m$ is the slope of the line, and
• $b$ is the y-intercept.

Step 1: Calculate the necessary sums

We need the following sums:

1. $\sum x$
2. $\sum y$
3. $\sum x^2$
4. $\sum xy$

Let's compute these:

$x = [4, 5, 6, 7, 8, 9, 10, 11]$ $y = [33.3, 34.4, 29.2, 27.1, 25.9, 25.6, 20.3, 19.4]$

Step 2: Calculate the sums

I'll calculate the sums of $x$, $y$, $x^2$, and $xy$ first.

$\sum x = 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 60$ $\sum y = 33.3 + 34.4 + 29.2 + 27.1 + 25.9 + 25.6 + 20.3 + 19.4 = 195.2$ $\sum x^2 = 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 = 16 + 25 + 36 + 49 + 64 + 81 + 100 + 121 = 492$ $\sum xy = (4 \times 33.3) + (5 \times 34.4) + (6 \times 29.2) + (7 \times 27.1) + (8 \times 25.9) + (9 \times 25.6) + (10 \times 20.3) + (11 \times 19.4)$ $\sum xy = 133.2 + 172 + 175.2 + 189.7 + 207.2 + 230.4 + 203 + 213.4 = 1524.1$

Step 3: Use the formulas for the slope $m$ and intercept $b$

The formulas for the slope $m$ and the intercept $b$ are:

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

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

Where:

• $n = 8$ (the number of data points).

Calculate the slope $m$:

$m = \frac{8 \times 1524.1 - (60 \times 195.2)}{8 \times 492 - 60^2}$ $m = \frac{12192.8 - 11712}{3936 - 3600}$ $m = \frac{480.8}{336}$ $m = 1.43$

Calculate the intercept $b$:

$b = \frac{195.2 - 1.43 \times 60}{8}$ $b = \frac{195.2 - 85.8}{8}$ $b = \frac{109.4}{8}$ $b = 13.68$

Final Regression Equation:

$y = 1.43x + 13.68$

So the regression line is:

$y = 1.43x + 13.68$

Let me know if you'd like further details or clarification on any step!

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Related Questions:

1. How would the regression line change if the data were normalized?
2. What is the interpretation of the slope and intercept in the context of the data?
3. How can the coefficient of determination (R-squared) be computed for this regression?
4. What other methods can be used to assess the accuracy of the regression model?
5. How would you predict the temperature for a year, say 12, using this model?

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Tip: When performing linear regression, always check the residuals (the differences between observed and predicted values) to ensure that the model fits the data well!