The scores of students in a national level examination are normally distributed with a mean of and a standard deviation of . If the value of the cumulative distribution of the standard normal random variable at is (i.e., ), then the probability that a randomly selected student scored between and is:

0.191 Explanation: Solution Step 1: Identify Parameters Mean ( ) = Standard Deviation ( ) = We need to find . Step 2: Convert to Standard Normal Variable ( ) The -score formula is: For : For : So, . Step 3: Use Properties of Normal Distribution By symmetry of the normal curve: We can express as: Step 4: Substitute Values Given: For a standard normal distribution, the area to the left of the mean ( ) is always : Therefore: Final Answer: The probability is 0.191 . (Option A)

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NIMCET 2025 — Mathematics PYQ

NIMCET | Mathematics | 2025

The scores of students in a national level examination are normally distributed with a mean of $500$ and a standard deviation of $100$ . If the value of the cumulative distribution of the standard normal random variable at $0.5$ is $0.691$ (i.e., $\Phi(0.5) = 0.691$ ), then the probability that a randomly selected student scored between $450$ and $500$ is:

NIMCET 2025 — Mathematics PYQ

Choose the correct answer:

Explanation

Solution

Step 1: Identify Parameters

Step 2: Convert to Standard Normal Variable ( $Z$ )

Step 3: Use Properties of Normal Distribution

Step 4: Substitute Values