http://www.ck12.org Chapter 5. Applications of Definite Integrals
f( 9 ) =( 0. 1 )^1 √ 2 πe−(^9 −^10.^2 )^2 /(^2 (^0.^1 )^2 )dx
= 1. 35 × 10 −^32 ,getting the area between 9 and 10 will yield a fairly good answer. Integrating numerically, we get
P( 9 ≤x≤ 10 ) =∫ 10
91
( 0. 1 )√ 2 πe−(x− 10. 2 )^2 /( 2 ( 0. 1 )^2 )dxP( 9 ≤x≤ 10. 2 )≈ 0. 02275
= 2 .28%,which says that we would expect 2.28% of the boxes to weigh less than 10 ounces.
- Theoretically the probability here will be exactly zero because we will be integrating from 10 to 10,which is zero.
However, since all scales have some error (call itε), practically we would find the probability that the weight falls
between 10−εand 10+ε.
Example 8:
An Intelligence Quotient or IQ is a score derived from different standardized tests attempting to measure the level
of intelligence of an adult human being. The average score of the test is 100 and the standard deviation is 15.- What is the percentage of the population that has a score between 85 and 115?
- What percentage of the population has a score above 140?
Solution:
- Using the normal probability density function,
f(x) =σ√^12 πe−(x−μ)^2 /(^2 σ^2 ),and substitutingμ=100 andσ= 15 ,
f(x) = 15 √^12 πe−(x−^100 )^2 /(^2 (^15 )^2 ).The percentage of the population that has a score between 85 and 115 is
P( 85 ≤x≤ 115 ) =∫ 115
851
15
√
2 πe−(x− 100 )^2 /( 2 ( 15 )^2 ).Again, the integral ofe−x^2 does not have an elementary anti-derivative and therefore cannot be evaluated. Using the
programing feature of a scientific calculator or a mathematical computer software, we get
∫ 115
851
15 √ 2 πe−(x− 100 )^2 /( 2 ( 15 )^2 )dx≈ 0. 68.