Scores of an IQ test have a bell-shaped distribution with a mean of 100 and a standard deviation of 19. Use the empirical rule to determine the following.
(a) What percentage of people has an IQ score between 62 and 138?
(b) What percentage of people has an IQ score less than 62 or greater than 138?
(c) What percentage of people has an IQ score greater than 119?
Given,
= 100 , = 19
a)
z-score = (X - ) /
For x = 62
z = ( 62 - 100) / 19
= -2
62 is 2 standard deviation below the mean.
For x = 138
z = ( 138 - 100) / 19
= 2
138 is 2 standard deviation above the mean.
Using empirical ( 68 - 95 - 99.7) rule,
Approximately 95% of the data falls in 2 standard deviation of the mean
P(62 < X < 138) = 95%
b)
P(X < 62 OR X > 138) = 1 - P(62 < X < 138)
= 1 - 0.95
= 0.05
= 5%
c)
z = ( 119 - 100) / 19
= 1
119 is 1 standard deviation above the mean.
Using empirical ( 68 - 95 - 99.7) rule,
Approximately 68% of the data falls in 1 standard deviation of the mean
That is ( 1 - 0.68) = 0.32 of the data falls outside the 1 standard deviation.
This means 16% data falls to either side of outside region of the 1 standard deviation.
P(X > 119) = 16%
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