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Step-by-Step Solution
Step 1: Understand the original data
We have a total of 2n observations. Half of them (n observations) are equal to $a$, and the other half (n observations) are equal to $-a$. Symbolically, the data set looks like:
$(\underbrace{a, a, \dots, a}_{n\text{ times}}, \underbrace{-a, -a, \dots, -a}_{n\text{ times}})$
Step 2: Find the mean of the original data
The sum of all observations in the original data is:
$$
\sum_{i=1}^{2n} x_i = n \cdot a + n \cdot (-a) = na - na = 0.
$$
Hence, the mean of the original data is:
$$
\overline{x} = \frac{\sum_{i=1}^{2n} x_i}{2n} = \frac{0}{2n} = 0.
$$
Step 3: Effect of adding a constant b on the mean
We add a constant $b$ to each observation. If $\overline{x}_{\text{new}}$ is the new mean, then:
$$
\overline{x}_{\text{new}} = \overline{x} + b.
$$
It is given that the new mean is 5, so:
$$
5 = 0 + b \quad \Longrightarrow \quad b = 5.
$$
Step 4: Effect of adding a constant b on the standard deviation
Adding a constant to every observation shifts the data but does not affect the spread. Therefore, the standard deviation remains the same before and after adding $b$.
Let $\sigma$ be the original standard deviation of the data. After adding $b$, the new standard deviation is still the same as the original standard deviation, and it is given to be 20.
Step 5: Compute the original standard deviation
Since the original mean is 0, we can write the original standard deviation $\sigma$ as:
$$
\sigma = \sqrt{\frac{1}{2n} \sum_{i=1}^{2n} (x_i - \overline{x})^2}
= \sqrt{\frac{1}{2n} \sum_{i=1}^{2n} x_i^2}.
$$
We know that $n$ of the $x_i$'s are $a$ and $n$ of them are $-a$. Thus, the sum of squares becomes:
$$
\sum_{i=1}^{2n} x_i^2 = n \cdot a^2 + n \cdot (-a)^2 = na^2 + na^2 = 2n a^2.
$$
Hence,
$$
\sigma = \sqrt{\frac{2n a^2}{2n}} = \sqrt{a^2} = |a|.
$$
Given $\sigma = 20$, we have:
$$
|a| = 20 \quad \Longrightarrow \quad a = 20 \quad (\text{taking the positive value for simplicity}).
$$
Step 6: Calculate $a^2 + b^2$
We have $a = 20$ and $b = 5$. Therefore:
$$
a^2 + b^2 = 20^2 + 5^2 = 400 + 25 = 425.
$$
Final Answer
The value of $a^2 + b^2$ is 425.
