Standard Deviation-Algebra 2 help!!!

Question

14.) Over 350 students took a college calculus final exam. The scores of the students follow a normal distribution. Using the information given below, determine the mean and the standard deviation for the students' scores. Give your answers to nearest tenth of a percent.
- 32% of the students scored either less than 63.3% or more than 85.9%.
- Only the top 2.5% of students scored higher than a 97%on the exam.
Mean:_______________________ Standard Deviation:____________________________

Answer 1

Based on the given information we have P(X < 63.3 or X > 85.9) = 0.32.

If that's the case then P(63.3 < X < 85.9) = 0.68. This is the probability that students scored between 63.3 and 85.9% on the exam.

We want to use the definition of standard deviation. 68% of the values are within 1 standard deviation of the mean. The mean is halfway between 63.3 and 85.9.

Mean = (63.3 + 85.9) / 2 = 74.6

To find what 1 standard deviation is, subtract 63.3 from 74.6 to get 11.3.

1 standard deviation: (63.3, 85.9) ⇒ 68%

2 standard deviations: (52, 97.2) ⇒ 95%

3 standard deviations: (40.7, 108.5) ⇒ 99.7%

Both conditions are met. Therefore, the normal distribution has the mean of 74.6% and the standard deviation of 11.3%.