Step 1: Convert scores to Z-scores

Use the formula: $$ Z = \frac{X - \mu}{\sigma} $$

For \( X = 500 \): $$ Z = \frac{500 - 500}{100} = 0 $$ For \( X = 450 \): $$ Z = \frac{450 - 500}{100} = -0.5 $$

Step 2: Find the probability using cumulative values

We calculate: $$ P(450 \leq X \leq 500) = P(-0.5 \leq Z \leq 0) = P(Z \leq 0) - P(Z \leq -0.5) $$

From symmetry: $$ P(Z \leq -0.5) = 1 - P(Z \leq 0.5) = 1 - 0.691 = 0.309 $$ and $$ P(Z \leq 0) = 0.5 $$

Step 3: Final Calculation

$$ P(-0.5 \leq Z \leq 0) = 0.5 - 0.309 = \boxed{0.191} $$

✅ Final Answer:

The probability that a randomly selected student scored between 450 and 500 is: 0.191