1. Original data set

   There are $2n$ observations:

   • $n$ observations are equal to $a$
   • $n$ observations are equal to $-a$

   So the original set is: $a,a,\dots,a,-a,-a,\dots,-a$

2. Mean of the original set

   Since half are $a$ and half are $-a$, $\bar{x}=\frac{na+n(-a)}{2n}=0$

3. Effect of adding constant $b$ to each observation

   New observations become: $a+b \quad \text{and} \quad -a+b$

   Adding a constant increases the mean by that constant, but does not change the standard deviation.

4. Using the new mean

   New mean is given as $5$.

   Since original mean was $0$, $0+b=5$ $b=5$

5. Using the new standard deviation

   New standard deviation is given as $20$.

   Since adding a constant does not change standard deviation, original standard deviation is also $20$.

6. Find original standard deviation

   Original mean is $0$. So variance is: $\sigma^2=\frac{n(a-0)^2+n(-a-0)^2}{2n}$ $=\frac{na^2+na^2}{2n}=a^2$

   Hence standard deviation is: $\sigma=|a|$

   Given $\sigma=20$, $|a|=20 \Rightarrow a^2=400$

7. Compute $a^2+b^2$

   $a^2+b^2=400+25=425$

8. Option check

   $425$ matches Option A.

Comparison with stored answer:

Stored correct answer is A, and our derived answer is also A.