The image contains three mathematical problems. I'll transcribe and solve each one:

Problem 1:

Find the least squares solution for the linear system that fits the following data points and find the slope $\hat{\beta}_1$ and the intercept $\hat{\beta}_0$ of the fitted line.

Data points:
$(0, 3), (1, 1), (2, 0), (3, 0), (4, -2), (5, -4)$

Problem 2:

Find the quadratic polynomial that best fits the following data under the least squares criterion:

$\{(1, 1), (2, 0), (3, 0), (4, 3)\}$

Problem 3:

Find the least squares polynomial of degrees 2 and 3 for the data in the following table. Compute the error in each case. Graph the data and the polynomials.

Data points:

Solving Problem 1:

We are tasked with finding the least squares solution for the linear system, fitting a straight line of the form $y = \hat{\beta}_0 + \hat{\beta}_1 x$.

The data points are:

We will calculate $\hat{\beta}_1$ (slope) and $\hat{\beta}_0$ (intercept) using the formulas for the least squares line:

$\hat{\beta}_1 = \frac{n\sum(x_i y_i) - \sum x_i \sum y_i}{n\sum(x_i^2) - (\sum x_i)^2}$ $\hat{\beta}_0 = \frac{\sum y_i - \hat{\beta}_1 \sum x_i}{n}$

Let me compute these values.For Problem 1, the least squares solution gives the following line equation:

$\hat{y} = 2.81 - 1.26x$

Thus, the slope $\hat{\beta}_1 = -1.26$ and the intercept $\hat{\beta}_0 = 2.81$.

I'll proceed with Problem 2, where we need to fit a quadratic polynomial to the given points:

$\{(1, 1), (2, 0), (3, 0), (4, 3)\}$

This involves solving for a quadratic equation of the form:

$y = ax^2 + bx + c$

Let me calculate the coefficients $a$, $b$, and $c$ using the least squares method.For Problem 2, the quadratic polynomial that best fits the data is:

$y = 1.00x^2 - 4.40x + 4.50$

Now, I'll move to Problem 3, where we need to fit both a degree 2 and degree 3 polynomial to the given data and compute the errors. The data points are:

Let me first compute the degree 2 polynomial.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can guide you through the process of solving the third problem manually. Let me know how you'd like to proceed!

Would you like any more details about the solutions so far? Here are some follow-up questions for further exploration:

1. How do you interpret the slope and intercept in a real-world context?
2. What is the significance of the least squares method in statistics?
3. Can you find the residuals for each data point for Problem 1?
4. How would adding more data points affect the least squares solution?
5. What alternative methods are there to fit curves other than least squares?

Tip: Always visualize the data and the fitted curve to ensure a proper fit and better understanding of the solution.