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Matakuliah Tahun Versi : I0134 – Metoda Statistika : 2005 : Revisi Pertemuan 15 Aplikasi Sebaran Normal 1 Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : • Mahasiswa dapat menggunakan tabel normal dan menghitung peluang Binom dengan pendekatan normal. 2 Outline Materi • Kaidah Peluang normal • Ketaksamaan Chebyshev • Pendekatan normal pada Binomial 3 The Normal Distribution as an Approximation to Other Probability Distributions The normal distribution with = 5.5 and = 1.6583 is a closer approximation to the binomial with n = 11 and p = 0.50. 4 Hampiran Normal P(x<4.5) = 0.2732 Normal Distribution: = 5.5, = 1.6583 Binomial Distribution: n = 11, p = 0.50 P(x4) = 0.2744 0.3 0.2 f(x) P(x) 0.2 0.1 0.1 0.0 0.0 0 5 10 0 1 2 3 4 Normal with mean = 5.50000 and standard deviation = 1.65830 x P( X <= x) 4.5000 0.2732 6 7 8 9 10 11 X X MTB > cdf 4.5; SUBC> normal 5.5 1.6583. Cumulative Distribution Function 5 MTB > cdf 4; SUBC> binomial 11,.5. Cumulative Distribution Function Binomial with n = 11 and p = 0.500000 x P( X <= x) 4.00 0.2744 5 Approximating a Binomial Probability Using the Normal Distribution b np a np P( a X b) P Z np(1 p) np(1 p) for n large (n 50) and p not too close to 0 or 1.00 6 Approximating a Binomial Probability Using the Normal Distribution b 0.5 np a 0.5 np P(a X b) P Z np(1 p) np(1 p) for n moderately large (20 n < 50). If p is either small (close to 0) or large (close to 1), use the Poisson approximation. 7 Using the Normal Transformation Example 4-1 X~N(160,302) Example 4-2 X~N(127,222) P (100 X 180) 100 X 180 P P ( X 150) X 150 P 100 160 180 160 P Z 30 30 P 2 Z .6667 0.4772 0.2475 0.7247 150 127 P Z 22 P Z 1.045 0.5 0.3520 0.8520 8 Using the Normal Transformation - Minitab Solutions MTB > cdf 100; SUBC> normal 160,30. Cumulative Distribution Function Normal with mean = 160.000 and standard deviation = 30.0000 x P( X <= x) 100.0000 0.0228 MTB > cdf 180; SUBC> normal 160,30. Cumulative Distribution Function Normal with mean = 160.000 and standard deviation = 30.0000 MTB > cdf 150; SUBC> normal 127,22. Cumulative Distribution Function Normal with = 127.000 and = 22.0000 x P( X <= x) 150.0000 0.8521 x P( X <= x) 180.0000 0.7475 9 Using the Normal Transformation Normal Distribution: = 383, = 12 Example 4-3 X~N(383,122) 0.05 P ( 394 X 399) 394 X 399 P P 0.9166 Z 1.333 0.4088 0.3203 0.0885 f(X) 0.03 0.02 0.01 Standard Normal Distribution 0.00 340 0.4 390 440 X 0.3 f(z) 399 383 394 383 P Z 12 12 0.04 0.2 0.1 0.0 -5 -4 -3 -2 -1 0 1 2 3 4 5 Z MTB > cdf 394; SUBC> normal 383,12. MTB > cdf 399; SUBC> normal 383,12. Cumulative Distribution Function Cumulative Distribution Function Normal with mean = 383.000 and standard deviation = 12.0000 Normal with mean = 383.000 and standard deviation = 12.0000 x P( X <= x) 394.0000 0.8203 x P( X <= x) 399.0000 0.9088 10 Normal Probabilities S ta n d a rd N o rm a l D is trib utio n • The probability that a normal • • 0.4 0.3 f(z) random variable will be within 1 standard deviation from its mean (on either side) is 0.6826, or approximately 0.68. The probability that a normal random variable will be within 2 standard deviations from its mean is 0.9544, or approximately 0.95. The probability that a normal random variable will be within 3 standard deviation from its mean is 0.9974. 0.2 0.1 0.0 -5 -4 -3 -2 -1 0 1 2 3 4 5 Z 11 • Selamat Belajar Semoga Sukses. 12