Probability of Students Scoring Below a Given Score

A group of 60 students took a mathematics exam. The scores on the exam were normally distributed with a mean score of 75 and a standard deviation of 10. Assuming the distribution of scores follows a normal curve, determine the proportion of students who scored below 65.

To find this, use the Z-score formula: $$ Z = \frac{X - \mu}{\sigma} $$

where:

• $X$ is the score of interest (65 in this case),
• $\mu$ is the mean (75),
• $\sigma$ is the standard deviation (10).

Using the Z-score, look up the score in the standard normal distribution table or use a calculator to find the corresponding percentile.