A certain data set has a mean of 27 and a standard deviation of 3.

Question

1.    A certain data set has a mean of 27 and a standard deviation of 3.

1. What proportion of the distribution would be found between 21 and 33?
2. What proportion would be between 18 and 21?
3. What proportion would be greater than 18?
4. What proportion would be between 24 and 33?

Answer 1

a) 95.5%. By the empirical rule 95% fall within 2 standard deviations from the mean. A more precise empirical rule is 68.3 - 95.5 - 99.7 for 1,2 & 3 standard deviations. Or use a z table to get 95.44%

b) 2.1% Take the difference between 99.7 and 95.5 = 4.2, divide by 2

or use z tables to get 2.15% take the difference of .0228 and 0.0013, what you find on the z tables for z=-2 and z=-3

c) (1-99.7)/2 = 0.15% by the empirical rule or by z tables 0.13%

d) 24 to 33 is 97.6% by the empirical rule Or 97.59% with z tables. It's within 2 standard deviations above the mean and 1 standard deviation below the mean

average 99.7 and 95.5 equals 97.6

The empirical rule is close enough and doesn't require tables or electronic calculators, just remembering 68.3- 95.5 - 99.7