To formulate the optimization problem for Dr. X, we need to minimize the sum of the squared errors between the true weight $w$ and the measurements from the 10 different scales.

Variables:

Let the true weight of the object be denoted by $w$ (which is the value Dr. X wants to estimate).

Measurements:

The measurements from the 10 different scales are:

$\text{Measurements} = \{100.8, 99.4, 101.3, 97.6, 102.5, 102.4, 104.6, 102.6, 95.1, 96.6\}$

These measurements are the observed weights, which are all subject to some error when compared to the true weight.

Measurement Error:

The error for each scale $i$ is defined as the difference between the measured value and the true weight $w$. This error is denoted by:

$\text{Error}_i = \text{Measurement}_i - w$

Sum of Squared Errors:

The total error that Dr. X wants to minimize is the sum of the squares of all these individual errors. So, the objective function $E(w)$ to minimize is:

$E(w) = \sum_{i=1}^{10} (\text{Measurement}_i - w)^2$

Substituting the values of the measurements:

$E(w) = (100.8 - w)^2 + (99.4 - w)^2 + (101.3 - w)^2 + (97.6 - w)^2 + (102.5 - w)^2 + (102.4 - w)^2 + (104.6 - w)^2 + (102.6 - w)^2 + (95.1 - w)^2 + (96.6 - w)^2$

Objective:

Dr. X's objective is to find the value of $w$ (the true weight) that minimizes this sum of squared errors $E(w)$. This is the optimization problem:

$\text{Minimize } E(w) = \sum_{i=1}^{10} (\text{Measurement}_i - w)^2$

Solving the Optimization Problem:

The solution to this problem can be found by taking the derivative of $E(w)$ with respect to $w$, setting it equal to zero, and solving for $w$. This will give the value of $w$ that minimizes the total error.

Would you like to go through the steps of solving this optimization problem analytically?

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Here are some related questions to further explore this topic:

1. How can we interpret the sum of squared errors in terms of statistical methods?
2. What is the relationship between the sum of squared errors and the concept of least squares regression?
3. How does the optimization problem change if we introduce weights for each measurement?
4. What role does the standard deviation play in estimating the true weight from the measurements?
5. How can this optimization framework be applied to more complex experiments with larger datasets?

Tip: In problems involving measurement errors, minimizing the sum of squared errors is a common approach, often leading to the method of least squares, which is fundamental in both statistics and machine learning.