Q.For the same distribution $X \sim N(50, 100)$, find $P(45 < X < 65)$.

This is a 'between two values' question, so both raw values must be standardized and the 'between' rule ($\Phi(z_2)-\Phi(z_1)$) applied.

Step 1 — Standardize both values. With $\mu=50,\sigma=10$:

Step 2 — Apply the symmetry rule for the negative z. From Section 4, $\Phi(-0.5) = 1-\Phi(0.5) = 1-0.6915 = 0.3085$.

Step 3 — Apply the 'between' rule.

Independent second method (dual-solve check): split the interval at the mean and add the two halves separately. The area from $x=45$ to $\mu=50$ equals the area from $z=-0.5$ to $z=0$, which by symmetry equals the area from $z=0$ to $z=0.5$, i.e. $\Phi(0.5)-0.5 = 0.6915-0.5 = 0.1915$. The area from $\mu=50$ to $x=65$ equals the area from $z=0$ to $z=1.5$, i.e. $\Phi(1.5)-0.5 = 0.9332-0.5=0.4332$. Adding the two halves: $0.1915+0.4332 = 0.6247$ — exactly matching Step 3, confirming the answer.

Forgetting to convert Φ(z) for a negative z using the symmetry rule (mistakenly looking up 'Φ(-0.5)' as if the table had negative entries, or wrongly using 1−Φ(0.5) as the FINAL answer instead of only the piece needed for Φ(−0.5)).