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Step-by-Step Solution
Step 1: Understand the Information Given
We have three observations: a, b, and c, such that
$b = a + c$. We also know that the standard deviation of the observations
a + 2, b + 2, and c + 2 is d. Because adding a constant
does not change the standard deviation, the standard deviation of
a, b, c is also d.
Step 2: Express the Mean of a, b, and c
The mean of a, b, c is given by:
$$
\overline{x} = \frac{a + b + c}{3}.
$$
Since $b = a + c$,
$$
a + b + c = a + (a + c) + c = 2(a + c).
$$
But because $b = a + c$, we get
$$
a + b + c = 2b \quad \Longrightarrow \quad \overline{x} = \frac{2b}{3}.
$$
Step 3: Use the Formula for the Standard Deviation
The standard deviation squared (variance) for three observations is:
$$
d^2 = \frac{a^2 + b^2 + c^2}{3} \;-\; \overline{x}^2.
$$
Substituting $\overline{x} = \frac{2b}{3}$ gives:
$$
d^2 = \frac{a^2 + b^2 + c^2}{3} \;-\; \left(\frac{2b}{3}\right)^2.
$$
Simplify the second term:
$$
\left(\frac{2b}{3}\right)^2 = \frac{4b^2}{9}.
$$
Hence,
$$
d^2 = \frac{a^2 + b^2 + c^2}{3} - \frac{4b^2}{9}.
$$
Step 4: Rearrange to Solve for b²
Multiply both sides by 9 to clear denominators:
$$
9d^2 = 3(a^2 + b^2 + c^2) - 4b^2.
$$
Rearrange to isolate $b^2$:
$$
3(a^2 + b^2 + c^2) - 4b^2 - 9d^2 = 0
$$
$$
3(a^2 + c^2) + 3b^2 - 4b^2 - 9d^2 = 0
$$
$$
3(a^2 + c^2) - b^2 - 9d^2 = 0
$$
$$
b^2 = 3(a^2 + c^2) - 9d^2.
$$
This matches the given correct option
$$
b^2 = 3(a^2 + c^2)\;-\;9d^2,
$$
completing the proof.
Answer
The correct statement is
$$
b^2 = 3(a^2 + c^2) - 9d^2.
$$
