14 slide no 28 chapter 10 using data ence 627

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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 Chi-square 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 chi-square 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 Chi-square 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 Chi-square 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: BestFit a...
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