Question

Scores of an IQ test have a​ bell-shaped distribution with a mean of 100 and a standard deviation of 17. Use the empirical rule to determine the following. ​(a) What percentage of people has an IQ score between 49 and 151​? ​(b) What percentage of people has an IQ score less than 83 or greater than 117​? ​(c) What percentage of people has an IQ score greater than 151​?

Answer 1

according to the emperical rule, 68% of data falls within the first standard deviation from the mean. 95% fall within two standard deviations. and 99.7% falls within three standard deviations of the mean.
here, mean = 100 and sd = 17
hence,
68% of data falls within (100-17, 100+17) = (83, 117).
95% fall within (100-2*17, 100+2*17) = (66,134).
99.7% falls within (100-3*17, 100+3*17) = (49, 151).

a)
Since, 99.7% falls within (100-3*17, 100+3*17) = (49, 151).
Hence, the percentage of people has an IQ score between 49 and 151 is 99.7%

b)
Since 68% of data falls within (100-17, 100+17) = (83, 117).
Hence, (100%-68%) = 32% of people has an IQ score less than 83 or greater than 117 .

c)
Since, 99.7% falls within (100-3*17, 100+3*17) = (49, 151).
(100%-99.7%) = 0.3% of the data falls outside (49, 151).
Since the normal distribution is symmetric about the mean, I can say that 0.15% of the data falls greater than 151.
hence 0.15% of people has an IQ score greater than 151