This part of the problem asks you to identify the expression that is minimized using the least squares method.

The least squares method involves minimizing the sum of the squared differences between the observed values $y_i$ and the predicted values from the model $\hat{y}_i$.

Model:

The model given is $y = c_0 + A_0 \sin(2t)$.

General Least Squares Formula:

The expression minimized in the least squares method is generally: $\sum_{i=1}^n \left( \text{Observed value} - \text{Model prediction} \right)^2$ In this context, the observed value is $y_i$, and the model prediction is $c_0 + A_0 \sin(2t_i)$.

Expression in the Problem:

The correct expression that corresponds to this method is: $\sum_{i=1}^5 \left( c_0 + A_0 \sin(2t_i) - y_i \right)^2$ This is the expression where the square of the difference between the observed data points $y_i$ and the model's prediction $c_0 + A_0 \sin(2t_i)$ is summed over all data points.

Answer:

The correct answer is the option: $\boxed{\sum_{i=1}^5 \left( c_0 + A_0 \sin(2t_i) - y_i \right)^2}$

Would you like more information on how this expression is derived or have any other questions?

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5 Related Questions:

1. How does the least squares method handle outliers in data?
2. Can the least squares method be used for models other than linear or sinusoidal?
3. How do you interpret the coefficients $c_0$ and $A_0$ after fitting the model?
4. What is the geometric interpretation of the least squares method?
5. How would you apply the least squares method if $t$ had different units or was measured on a different scale?

Tip: In regression, always ensure that the function you are minimizing corresponds correctly to the difference between observed and predicted values, squared and summed over all data points.