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Pertemuan 24 Uji Kebaikan Suai

Pertemuan 24 Uji Kebaikan Suai. Matakuliah : I0134 – Metoda Statistika Tahun : 2005 Versi : Revisi. Learning Outcomes. Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : Mahasiswa dapat menghasilkan uji suatu sebaran (distribusi). Outline Materi. Statistik uji khi-kuadrat

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Pertemuan 24 Uji Kebaikan Suai

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  1. Pertemuan 24Uji Kebaikan Suai Matakuliah : I0134 – Metoda Statistika Tahun : 2005 Versi : Revisi

  2. Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : • Mahasiswa dapat menghasilkan uji suatu sebaran (distribusi).

  3. Outline Materi • Statistik uji khi-kuadrat • Uji sebaran seragam • Uji sebaran binomial • Uji sebaran normal

  4. Tests of Goodness of Fit and Independence • Goodness of Fit Test: A Multinomial Population • Tests of Independence: Contingency Tables • Goodness of Fit Test: Poisson and Normal Distributions

  5. Goodness of Fit Test:A Multinomial Population 1. Set up the null and alternative hypotheses. 2. Select a random sample and record the observed frequency, fi , for each of the k categories. 3. Assuming H0 is true, compute the expected frequency, ei , in each category by multiplying the category probability by the sample size. continued

  6. Goodness of Fit Test:A Multinomial Population 4. Compute the value of the test statistic. 5. Reject H0 if (where  is the significance level and there are k - 1 degrees of freedom).

  7. Contoh Soal: Finger Lakes Homes • Multinomial Distribution Goodness of Fit Test The number of homes sold of each model for 100 sales over the past two years is shown below. Model Colonial Ranch Split-Level A-Frame # Sold 30 20 35 15

  8. Contoh Soal: Finger Lakes Homes • Multinomial Distribution Goodness of Fit Test • Notation pC = popul. proportion that purchase a colonial pR = popul. proportion that purchase a ranch pS = popul. proportion that purchase a split-level pA = popul. proportion that purchase an A-frame • Hypotheses H0: pC = pR = pS = pA = .25 Ha: The population proportions are not pC = .25, pR = .25, pS = .25, and pA = .25

  9. Contoh Soal: Finger Lakes Homes • Multinomial Distribution Goodness of Fit Test • Expected Frequencies e1 = .25(100) = 25 e2 = .25(100) = 25 e3 = .25(100) = 25 e4 = .25(100) = 25 • Test Statistic = 1 + 1 + 4 + 4 = 10

  10. Contoh Soal: Finger Lakes Homes • Multinomial Distribution Goodness of Fit Test • Rejection Rule With  = .05 and k - 1 = 4 - 1 = 3 degrees of freedom Do Not Reject H0 Reject H0 2 7.81

  11. Contoh Soal: Finger Lakes Homes • Multinomial Distribution Goodness of Fit Test • Conclusion c2 = 10 > 7.81, so we reject the assumption there is no home style preference, at the .05 level of significance.

  12. Goodness of Fit Test: Poisson Distribution 1. Set up the null and alternative hypotheses. 2. Select a random sample and a. Record the observed frequency, fi , for each of the k values of the Poisson random variable. b. Compute the mean number of occurrences, . 3. Compute the expected frequency of occurrences, ei , for each value of the Poisson random variable.continued

  13. Goodness of Fit Test: Poisson Distribution 4. Compute the value of the test statistic. 5. Reject H0 if (where  is the significance level and there are k - 2 degrees of freedom).

  14. Contoh Soal: Troy Parking Garage • Poisson Distribution Goodness of Fit Test In studying the need for an additional entrance to a city parking garage, a consultant has recommended an approach that is applicable only in situations where the number of cars entering during a specified time period follows a Poisson distribution.

  15. Contoh Soal: Troy Parking Garage • Poisson Distribution Goodness of Fit Test A random sample of 100 one-minute time intervals resulted in the customer arrivals listed below. A statistical test must be conducted to see if the assumption of a Poisson distribution is reasonable. # Arrivals0 1 2 3 4 5 6 7 8 9 10 11 12 Frequency 0 1 4 10 14 20 12 12 9 8 6 3 1

  16. Contoh Soal: Troy Parking Garage • Poisson Distribution Goodness of Fit Test • Hypotheses H0: Number of cars entering the garage during a one-minute interval is Poisson distributed. Ha: Number of cars entering the garage during a one-minute interval is not Poisson distributed

  17. Contoh Soal: Troy Parking Garage • Poisson Distribution Goodness of Fit Test • Estimate of Poisson Probability Function otal Arrivals = 0(0) + 1(1) + 2(4) + . . . + 12(1) = 600 Total Time Periods = 100 Estimate of  = 600/100 = 6 Hence,

  18. Contoh Soal: Troy Parking Garage • Poisson Distribution Goodness of Fit Test • Expected Frequencies xf (x ) xf (x ) xf (x ) xf (x ) 0 .0025 .25 7 .1389 13.89 1 .0149 1.49 8 .1041 10.41 2 .0446 4.46 9 .0694 6.94 3 .0892 8.92 10 .0417 4.17 4 .1339 13.39 11 .0227 2.27 5 .1620 16.20 12.01551.55 6 .1606 16.06 Total 1.0000 100.00

  19. Contoh Soal: Troy Parking Garage • Poisson Distribution Goodness of Fit Test • Observed and Expected Frequencies ifieifi - ei 0 or 1 or 2 5 6.20 -1.20 3 10 8.92 1.08 4 14 13.39 .61 5 20 16.20 3.80 6 12 16.06 -4.06 7 12 13.89 -1.89 8 9 10.41 -1.41 9 8 6.94 1.06 10 or more 10 7.99 2.01

  20. Contoh Soal: Troy Parking Garage • Poisson Distribution Goodness of Fit Test • Test Statistic • Rejection Rule With  = .05 and k - p - 1 = 9 - 1 - 1 = 7 d.f. (where k = number of categories and p = number of population parameters estimated), Reject H0 if 2 > 14.07 • Conclusion We cannot reject H0. There’s no reason to doubt the assumption of a Poisson distribution.

  21. Goodness of Fit Test: Normal Distribution 4. Compute the value of the test statistic. 5. Reject H0 if (where  is the significance level and there are k - 3 degrees of freedom).

  22. Contoh Soal: Victor Computers • Normal Distribution Goodness of Fit Test Victor Computers manufactures and sells a general purpose microcomputer. As part of a study to evaluate sales personnel, management wants to determine if the annual sales volume (number of units sold by a salesperson) follows a normal probability distribution.

  23. Contoh Soal: Victor Computers • Normal Distribution Goodness of Fit Test A simple random sample of 30 of the salespeople was taken and their numbers of units sold are below. 33 43 44 45 52 52 56 58 63 64 64 65 66 68 70 72 73 73 74 75 83 84 85 86 91 92 94 98 102 105 (mean = 71, standard deviation = 18.54)

  24. Contoh Soal: Victor Computers • Normal Distribution Goodness of Fit Test • Hypotheses H0: The population of number of units sold has a normal distribution with mean 71 and standard deviation 18.54. Ha: The population of number of units sold does not have a normal distribution with mean 71 and standard deviation 18.54.

  25. Contoh Soal: Victor Computers • Normal Distribution Goodness of Fit Test • Interval Definition To satisfy the requirement of an expected frequency of at least 5 in each interval we will divide the normal distribution into 30/5 = 6 equal probability intervals.

  26. Contoh Soal: Victor Computers • Normal Distribution Goodness of Fit Test • Interval Definition Areas = 1.00/6 = .1667 53.02 71 88.98 = 71 + .97(18.54) 63.03 78.97

  27. Contoh Soal: Victor Computers • Normal Distribution Goodness of Fit Test • Observed and Expected Frequencies ifieifi – ei Less than 53.02 6 5 1 53.02 to 63.03 3 5 -2 63.03 to 71.00 6 5 1 71.00 to 78.97 5 5 0 78.97 to 88.98 4 5 -1 More than 88.98 6 5 1 Total 30 30

  28. Victor Computers • Normal Distribution Goodness of Fit Test • Test Statistic • Rejection Rule With  = .05 and k - p - 1 = 6 - 2 - 1 = 3 d.f., Reject H0 if 2 > 7.81 • Conclusion We cannot reject H0. There is little evidence to support rejecting the assumption the population is normally distributed with  = 71 and  = 18.54.

  29. Selamat Belajar Semoga Sukses.

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