To determine the slope $\hat{\beta}_1$ and y-intercept $\hat{\beta}_0$ for the regression line, we use the following formulas:

$\hat{\beta}_1 = \frac{n(\sum x_i y_i) - (\sum x_i)(\sum y_i)}{n(\sum x_i^2) - (\sum x_i)^2}$

$\hat{\beta}_0 = \frac{(\sum y_i) - \hat{\beta}_1(\sum x_i)}{n}$

Given:

• $n = 12$
• $\sum x_i = 460.53$
• $\sum y_i = -6349.7$
• $\sum x_i^2 = 17875.1$
• $\sum y_i^2 = 3421892$ (not needed for these calculations)
• $\sum x_i y_i = -246677$

Calculate $\hat{\beta}_1$:

$\hat{\beta}_1 = \frac{12(-246677) - (460.53)(-6349.7)}{12(17875.1) - (460.53)^2}$

Calculate the numerator and the denominator separately, then solve for $\hat{\beta}_1$.

Calculate $\hat{\beta}_0$:

Plug $\hat{\beta}_1$ into the formula for $\hat{\beta}_0$:

$\hat{\beta}_0 = \frac{(-6349.7) - \hat{\beta}_1(460.53)}{12}$

Finally, estimate the true mean value of $Y$ when $x = 41$ using:

$\hat{Y} = \hat{\beta}_0 + \hat{\beta}_1(41)$

Let's calculate these values:

Calculation:

First, calculate $\hat{\beta}_1$:

1. Numerator: $12(-246677) - (460.53)(-6349.7) = -2960124 + 2930344.121 = -29779.879$

2. Denominator: $12(17875.1) - (460.53)^2 = 214501.2 - 212088.2809 = 2412.9191$

3. So, $\hat{\beta}_1 = \frac{-29779.879}{2412.9191} \approx -12.3444$

Now, calculate $\hat{\beta}_0$:

$\hat{\beta}_0 = \frac{(-6349.7) - (-12.3444)(460.53)}{12} = \frac{-6349.7 + 5684.732852}{12} = \frac{-664.967148}{12} \approx -55.4139$

Finally, estimate $Y$ for $x = 41$:

$\hat{Y} = -55.4139 + (-12.3444)(41) = -55.4139 - 506.1234 = -561.5373$

Answers:

$\hat{\beta}_1 \approx -12.3444$

$\hat{\beta}_0 \approx -55.4139$

Estimate when $x = 41$: $-561.54$