Unformatted text preview: from an assumed or
theoretical distribution. 14 Slide No. 28 CHAPTER 10. USING DATA ENCE 627 ©Assakkaf Testing the Validity of Assumed
Distribution Chisquare Test for Goodness of Fit
– The basis for the appraising the goodness
of the comparison is given by the following
k
test statistic:
(Oi − Ei )2
2
χ =∑
Ei
i =1
Where χ2 is the computed value of a random variable
having a chisquare distribution with k – 1 degrees of
freedom; Oi and Ei are the observed and expected
frequencies in cell (or interval) i, and k is the number
of discrete cells (intervals) into which data were
separated. Slide No. 29 CHAPTER 10. USING DATA ENCE 627 ©Assakkaf Testing the Validity of Assumed
Distribution
Chisquare Test for Goodness of Fit
– Degrees of Freedom • If the mean and standard deviation of the
sample are needed to compute the expected
frequencies, then two additional degrees of
freedom are subtracted (i.e., k – 3).
• If the mean and standard deviation are
obtained from past experience or other
sources, then the number of degrees of
freedom for the test statistic remains k – 1. 15 Slide No. 30 CHAPTER 10. USING DATA ENCE 627 ©Assakkaf Testing the Validity of Assumed
Distribution Chisquare Test for Goodness of Fit
– If the assumed distribution yields
k χ =∑
2 i =1 (Oi − Ei )2 < χ 2
Ei α, ν 1. The assumed theoretical distribution is an
acceptable model if χ2 < χ2α,ν
2. Otherwise, it is not acceptable at the α
significance level. CHAPTER 10. USING DATA Slide No. 31
ENCE 627 ©Assakkaf Testing the Validity of Assumed
Distribution
Example: Rainstorms Severe rainstorms have been recorded at a
given station over a period of 66 years. During
this period, there were 20 years without severe
rainstorms; and 23, 15, 6, and 2 years,
respectively, with 1, 2, 3, and 4 rainstorms
annually. Judging from the shape of the
histogram, a Poisson distribution seems an
appropriate model for the annual number of
rainstorms. Is this claim valid? Use a
significance level of 5%. 16 Slide No. 32 CHAPTER 10. USING DATA ENCE 627 ©Assakkaf Testing the Validity of Assumed
Distribution
Example (cont’d): Rainstorms Slide No. 33 CHAPTER 10. USING DATA ENCE 627 ©Assakkaf Testing the Validity of Assumed
Distribution
Example (cont’d): Rainstorms (Oi − Ei )2 No. of storms
at station per
year Observed
frequency, Oi Theoretical
frequency, Ei 0 20 19.94 0.0036 0.0002 1 23 23.87 0.7569 0.0317 2 15 14.29 0.5041 0.0353 >3 8 7.90 0.0100 0.0013 ∑ 99 66.00 (Oi − Ei ) 2 Ei 0.0685 17 Slide No. 34 CHAPTER 10. USING DATA ENCE 627 ©Assakkaf Testing the Validity of Assumed
Distribution
Example (cont’d): Rainstorms
α = 0.05 ⇒ 1  α = 1  0.05 = 0.95 X 23 + 2 × 15 + 3 × 6 + 4 × 2 79
=
=
= 1.197 rainstorms/year
t
66
66
From Chi  squares Table, for α = 0.05, and ν = k − 2 = 4 − 2 = 2,
λ= 2
χ 0.05,2 = 5.995 Since, k (Oi − Ei )2
2
∑
= 0.068 < χ α, ν = 5.995 i =1
Ei Hence, the Poisson distribution is a valid model at the 5% significance level. ( ) CHAPTER 10. USING DATA Slide No. 35
ENCE 627 ©Assakkaf Software for Fitting Distributions:
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This note was uploaded on 10/24/2012 for the course ESI 6385 taught by Professor Mansoorehmollaghasemi during the Fall '12 term at University of Central Florida.
 Fall '12
 MansoorehMollaghasemi

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