Using Normal Distribution Using Exponential Distribution. Examples 1. Artin Armagan. Sta February 3, Artin Armagan Examples 1

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1 Examples 1 Artin Armagan Sta. 113 February 3, 2009

2 Table of contents 1 Using Normal Distribution 2

3 IQ scores Using Normal Distribution An Intelligence Quotient or IQ is a score derived from one of several different standardized tests attempting to measure intelligence.

4 IQ scores Using Normal Distribution An Intelligence Quotient or IQ is a score derived from one of several different standardized tests attempting to measure intelligence. IQ scores are used in many contexts: as predictors of educational achievement or special needs, by social scientists who study the distribution of IQ scores in populations and the relationships between IQ score and other variables, and as predictors of job performance and income.

5 IQ scores Using Normal Distribution An Intelligence Quotient or IQ is a score derived from one of several different standardized tests attempting to measure intelligence. IQ scores are used in many contexts: as predictors of educational achievement or special needs, by social scientists who study the distribution of IQ scores in populations and the relationships between IQ score and other variables, and as predictors of job performance and income. Sometimes, the difficulty and number of questions on an IQ test is selected in order to yield normal distributed results. Or else, the raw test scores are converted to IQ values by fitting them to the normal distribution. In either case, it is the deliberate result of test construction or score interpretation that leads to IQ scores being normally distributed for the majority of the population. However, the question whether intelligence itself is normally distributed is more involved, because intelligence is a latent variable, therefore its distribution cannot be observed directly.

6 IQ scores Using Normal Distribution

7 IQ scores Using Normal Distribution Abraham Lincoln and Albert Einstein have estimated IQ scores of 128 and 160 respectively. What is the probability that a randomly chosen person has an IQ greater than Lincoln s and smaller than Einstein s?

8 IQ scores vs. fluoride levels in drinking water According to a recently released Chinese study, there is a clear relation between fluoride levels in the water that is consumed by a population and low Intelligence quotient scores of children who do the consuming. 512 children were tested, aged 8-13 years from Wamiao, a high-fluoride village, and Xinhuai, a low-fluoride village. The high-fluoride village was divided into five subgroups by water fluoride levels, from 0.62 mg Fluoride per Liter to 4.38 mg F/L. As water fluoride levels increased within the high fluoride village, IQ decreased. When the data from the 512 children in the two villages were considered as a whole, a significant inverse correlation was found between IQ and the level of fluoride in urine, the researchers report.

9 Approximating the Binomial Distribution P(X x) = B(x; n, p) ( area under the normal curve to the left of x ( ) x +.5 np = Φ npq )

10 Approximating the Binomial Distribution Chapter 4, question #50 in Devore. a. ( ) P(X 40) 1 Φ = b. ( ) P(X 50) Φ =

11 Exponential Distribution Chapter 4, question #63 in Devore. Cost functions: h 1 (x) = 10x { 99 0 < x 20 h 2 (x) = (x 20) x > 20 Eh(x) = h(x)p(x)dx.

12 Exponential Distribution Eh 2(x) = 99 Z 20 0 Eh 1(x) = 10 λe λx dx + Z = e 20λ + 10λ 0 Z 20 Z λe λx dx = 10 λ. 0 ( (x 20))λe λx dx xe λx dx {z } gamma function Z Z 20 10λ xe λx dx 0 {z } incomplete gamma function = e 20λ + 10 ue u du 10 ue u du λ 0 λ 0 {z } {z } = e 20λ + 10 λ u=λx Z 20λ 1 0 Z 20λ «ue u du u=λx

13 Exponential Distribution