Question

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, then the probability that a randomly selected student scored between 450 and 500 is:

• A. 0.191
• B. 0.391
• C. 0.591
• D. 0.091

Hint:

To find probabilities for normally distributed data, first convert the values to Z-scores and then use the standard normal distribution table.

Answer 1

The Correct Option is A

We are given that the mean score $\mu = 500$ and the standard deviation $\sigma = 100$. To find the probability that a student scored between 450 and 500, we first convert the scores to standard normal form using the Z-score formula: \[ Z = \frac{X - \mu}{\sigma} \] For $X = 450$: \[ Z_1 = \frac{450 - 500}{100} = -0.5 \] For $X = 500$: \[ Z_2 = \frac{500 - 500}{100} = 0 \] From the standard normal table, the cumulative probability for $Z = -0.5$ is 0.3085, and for $Z = 0$ is 0.5. The probability that a student scored between 450 and 500 is the difference between these two values: \[ P(450 \leq X \leq 500) = 0.5 - 0.3085 = 0.1915 \] Thus, the probability is approximately 0.191.