Statistical inferences question

Question

The scores made by candidates in a certain test are normally distributed with mean 500 and standard deviation 100.

What is the probability that a candidate received scores?

i. greater than 700

ii. Less than 400

iii. between 400 and 600

iv. between 300 and 400

v. between 600 and 700

Answer 1

i. 700 is 2 standard deviations above the mean. by the empirical rule 99.7% is within 2 standard deviations of the mean. the remaining 0.3% is divided in half, one half above 700 and one half below 300. 0.15% of the scores > 700. Probability of >700 is 0.15%

ii 16% half of 100-68 = 32/2 = 16

iii. 68%.

iv. 13.5% half of 95-68 = 27/2

v. 13.5%

the empirical rule is 68%-95%-99.7% for within 1, 2 and 3 standard deviations from the mean