Matakuliah Tahun Versi : A0064 / Statistik Ekonomi : 2005 : 1/1 Pertemuan 10 Sebaran Normal-2 1 Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : • Menghitung beberapa contoh permasalahan yang berkaitan dengan luas daerah di bawah kurva normal, transformasi variabel acak normal, dan pendekatan sebaran binomial dengan sebaran normal 2 Outline Materi • Transformasi Variabel Acak Normal • Pendekatan Sebaran Binomial dengan Sebaran Normal 3 COMPLETE 4-4 BUSINESS STATISTICS 5th edi tion 4-4 The Transformation of Normal Random Variables The area within k of the mean is the same for all normal random variables. So an area under any normal distribution is equivalent to an area under the standard normal. In this example: P(40 X P(-1 Z since m5and The transformation of X to Z: - m x X Z Normal Distribution: m =50, =10 x 0.07 0.06 Transformation f(x) (1) Subtraction: (X - mx) 0.05 0.04 0.03 =10 { 0.02 Standard Normal Distribution 0.01 0.00 0.4 0 10 30 40 50 60 70 80 90 100 X 0.3 0.2 (2) Division by x) { f(z) 20 1.0 0.1 X mx + Z x 0.0 -5 -4 -3 -2 -1 0 1 2 3 4 5 Z McGraw-Hill/Irwin The inverse transformation of Z to X: Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE 4-5 BUSINESS STATISTICS 5th edi tion Using the Normal Transformation Example 4-9 X~N(160,302) Example 4-10 X~N(127,222) P (100 X 180) 100 - m X - m 180 - m P P( X < 150) X - m < 150 - m P 100 - 160 180 - 160 P Z 30 30 ( P -2 Z .6667 0.4772 + 0.2475 0.7247 McGraw-Hill/Irwin 150 - 127 P Z < 22 ( P Z < 1.045 0.5 + 0.3520 0.8520 Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE 4-6 BUSINESS STATISTICS 5th edi tion Using the Normal Transformation Example 4-11 Normal Distribution: m = 383, = 12 Example 4-11 X~N(383,122) 0.05 P ( 394 X 399) 394 - m X - m 399 - m 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) 394 - 383 399 - 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 Template solution McGraw-Hill/Irwin Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE BUSINESS STATISTICS 4-7 5th edi tion The Transformation of Normal Random Variables The transformation of X to Z: Z The inverse transformation of Z to X: X - mx X m + Z x x x The transformation of X to Z, where a and b are numbers:: a - m < < P ( X a ) P Z b - m > > P ( X b) P Z b - m a-m < < < < P (a X b ) P Z McGraw-Hill/Irwin Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE 4-8 BUSINESS STATISTICS 5th edi tion Normal Probabilities (Empirical Rule) S ta n d a rd N o rm a l D is trib utio n • The probability that a normal • • McGraw-Hill/Irwin 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. Aczel/Sounderpandian 0.2 0.1 0.0 -5 -4 -3 -2 -1 0 1 2 3 4 5 Z © The McGraw-Hill Companies, Inc., 2002 COMPLETE BUSINESS STATISTICS 4-9 5th edi tion 4-5 The Inverse Transformation The area within k of the mean is the same for all normal random variables. To find a probability associated with any interval of values for any normal random variable, all that is needed is to express the interval in terms of numbers of standard deviations from the mean. That is the purpose of the standard normal transformation. If X~N(50,102), x - m 70 - m 70 - 50 > P Z > P( Z > 2) P( X > 70) P 10 That is, P(X >70) can be found easily because 70 is 2 standard deviations above the mean of X: 70 = m + 2. P(X > 70) is equivalent to P(Z > 2), an area under the standard normal distribution. Normal Distribution: m = 124, = 12 Example 4-12 X~N(124,122) P(X > x) = 0.10 and P(Z > 1.28) 0.10 x = m + z = 124 + (1.28)(12) = 139.36 0.04 0.03 . . . . . . . . . . . . . . . .07 . . . 0.3790 0.3980 0.4147 . . . McGraw-Hill/Irwin .08 . . . 0.3810 0.3997 0.4162 . . . .09 . . . 0.3830 0.4015 0.4177 . . . f(x) z . . . 1.1 1.2 1.3 . . . 0.02 0.01 0.01 0.00 80 130 X Aczel/Sounderpandian 180 139.36 © The McGraw-Hill Companies, Inc., 2002 COMPLETE BUSINESS STATISTICS 4-10 5th edi tion Template Solution for Example 4-12 Example 4-12 X~N(124,122) P(X > x) = 0.10 and P(Z > 1.28) 0.10 x = m + z = 124 + (1.28)(12) = 139.36 McGraw-Hill/Irwin Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE 4-11 BUSINESS STATISTICS 5th edi tion The Inverse Transformation (Continued) Example 4-14 X~N(2450,4002) P(a<X<b)=0.95 and P(-1.96<Z<1.96)0.95 x = m z = 2450 ± (1.96)(400) = 2450 ±784=(1666,3234) P(1666 < X < 3234) = 0.95 Example 4-13 X~N(5.7,0.52) P(X > x)=0.01 and P(Z > 2.33) 0.01 x = m + z = 5.7 + (2.33)(0.5) = 6.865 z . . . 2.2 2.3 2.4 . . . .02 . . . 0.4868 0.4898 0.4922 . . . . . . . . . . . . . . . . . . .03 . . . 0.4871 0.4901 0.4925 . . . .04 . . . 0.4875 0.4904 0.4927 . . . z . . . 1.8 1.9 2.0 . . Normal Distribution: m = 5.7 = 0.5 .06 . . . 0.4686 0.4750 0.4803 . . .07 . . . 0.4693 0.4756 0.4808 . . 0.0015 Area = 0.49 0.6 .4750 .4750 0.0010 f(x) 0.5 f(x) .05 . . . 0.4678 0.4744 0.4798 . . Normal Distribution: m = 2450 = 400 0.8 0.7 . . . . . . . . . . . . . . 0.4 X.01 = m+z = 5.7 + (2.33)(0.5) = 6.865 0.3 0.0005 0.2 .0250 .0250 Area = 0.01 0.1 0.0 0.0000 3.2 4.2 5.2 6.2 7.2 8.2 1000 2000 X -5 -4 -3 -2 -1 0 z McGraw-Hill/Irwin 3000 4000 X 1 2 3 4 5 -5 -4 -3 -2 -1.96 Z.01 = 2.33 Aczel/Sounderpandian -1 0 Z 1 2 3 4 5 1.96 © The McGraw-Hill Companies, Inc., 2002 COMPLETE 4-12 BUSINESS STATISTICS 5th edi tion Finding Values of a Normal Random Variable, Given a Probability Normal Distribution: m = 2450, = 400 0.0012 . 0.0010 . 0.0008 . f(x) 1. Draw pictures of the normal distribution in question and of the standard normal distribution. 0.0006 . 0.0004 . 0.0002 . 0.0000 1000 2000 3000 4000 X S tand ard Norm al D istrib utio n 0.4 f(z) 0.3 0.2 0.1 0.0 -5 -4 -3 -2 -1 0 1 2 3 4 5 Z McGraw-Hill/Irwin Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE 4-13 BUSINESS STATISTICS 5th edi tion Finding Values of a Normal Random Variable, Given a Probability Normal Distribution: m = 2450, = 400 0.0012 . .4750 0.0010 . .4750 0.0008 . f(x) 1. Draw pictures of the normal distribution in question and of the standard normal distribution. 0.0006 . 0.0004 . 0.0002 . .9500 0.0000 1000 2000 3000 4000 X S tand ard Norm al D istrib utio n 0.4 .4750 .4750 0.3 f(z) 2. Shade the area corresponding to the desired probability. 0.2 0.1 .9500 0.0 -5 -4 -3 -2 -1 0 1 2 3 4 5 Z McGraw-Hill/Irwin Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE 4-14 BUSINESS STATISTICS 5th edi tion Finding Values of a Normal Random Variable, Given a Probability Normal Distribution: m = 2450, = 400 3. From the table of the standard normal distribution, find the z value or values. 0.0012 . .4750 0.0010 . .4750 0.0008 . f(x) 1. Draw pictures of the normal distribution in question and of the standard normal distribution. 0.0006 . 0.0004 . 0.0002 . .9500 0.0000 1000 2000 3000 4000 X 2. Shade the area corresponding to the desired probability. S tand ard Norm al D istrib utio n 0.4 .4750 f(z) z . . . 1.8 1.9 2.0 . . . . . . . . . . . . . . . . .05 . . . 0.4678 0.4744 0.4798 . . McGraw-Hill/Irwin .06 . . . 0.4686 0.4750 0.4803 . . .4750 0.3 .07 . . . 0.4693 0.4756 0.4808 . . 0.2 0.1 .9500 0.0 -5 -4 -3 -2 -1 0 1 2 3 4 5 Z -1.96 Aczel/Sounderpandian 1.96 © The McGraw-Hill Companies, Inc., 2002 COMPLETE 4-15 BUSINESS STATISTICS 5th edi tion Finding Values of a Normal Random Variable, Given a Probability Normal Distribution: m = 2450, = 400 3. From the table of the standard normal distribution, find the z value or values. 0.0012 . .4750 0.0010 . .4750 0.0008 . f(x) 1. Draw pictures of the normal distribution in question and of the standard normal distribution. 0.0006 . 0.0004 . 0.0002 . .9500 0.0000 1000 2000 3000 4000 X 2. Shade the area corresponding to the desired probability. 0.4 .4750 . . . . . . . . . . . . . . .05 . . . 0.4678 0.4744 0.4798 . . McGraw-Hill/Irwin .06 . . . 0.4686 0.4750 0.4803 . . .4750 0.3 f(z) z . . . 1.8 1.9 2.0 . . 4. Use the transformation from z to x to get value(s) of the original random variable. S tand ard Norm al D istrib utio n .07 . . . 0.4693 0.4756 0.4808 . . 0.2 0.1 .9500 0.0 -5 -4 -3 -2 -1 0 1 2 Z -1.96 Aczel/Sounderpandian 3 4 5 x = m z = 2450 ± (1.96)(400) = 2450 ±784=(1666,3234) 1.96 © The McGraw-Hill Companies, Inc., 2002 COMPLETE 4-16 BUSINESS STATISTICS 5th edi tion Finding Values of a Normal Random Variable, Given a Probability The normal distribution with m = 3.5 and = 1.323 is a close approximation to the binomial with n = 7 and p = 0.50. P(x<4.5) = 0.7749 Normal Distribution: m = 3.5, = 1.323 Binomial Distribution: n = 7, p = 0.50 0.3 0.3 P( x 4) = 0.7734 0.2 f(x) P(x) 0.2 0.1 0.1 0.0 0.0 0 5 10 0 1 X 2 3 4 5 6 7 X MTB > cdf 4.5; SUBC> normal 3.5 1.323. Cumulative Distribution Function MTB > cdf 4; SUBC> binomial 7,.5. Cumulative Distribution Function Normal with mean = 3.50000 and standard deviation = 1.32300 Binomial with n = 7 and p = 0.500000 x 4.5000 McGraw-Hill/Irwin P( X <= x) 0.7751 Aczel/Sounderpandian x 4.00 P( X <= x) 0.7734 © The McGraw-Hill Companies, Inc., 2002 COMPLETE 4-17 BUSINESS STATISTICS 5th edi tion 4-6 The Normal Approximation of Binomial Distribution The normal distribution with m = 5.5 and = 1.6583 is a closer approximation to the binomial with n = 11 and p = 0.50. P(x < 4.5) = 0.2732 Normal Distribution: m = 5.5, = 1.6583 Binomial Distribution: n = 11, p = 0.50 P(x 4) = 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 4 5 6 7 8 9 10 11 MTB > cdf 4; SUBC> binomial 11,.5. Cumulative Distribution Function MTB > cdf 4.5; SUBC> normal 5.5 1.6583. Cumulative Distribution Function Normal with mean = 5.50000 and standard deviation = 1.65830 Binomial with n = 11 and p = 0.500000 x 4.00 P( X <= x) 0.2732 McGraw-Hill/Irwin 3 X X x 4.5000 2 Aczel/Sounderpandian P( X <= x) 0.2744 © The McGraw-Hill Companies, Inc., 2002 COMPLETE BUSINESS STATISTICS 4-18 5th edi tion Approximating a Binomial Probability Using the Normal Distribution a - np b - np Z P( a X b) & P np(1 - p) np(1 - p) for n large (n 50) and p not too close to 0 or 1.00 or: a - 0.5 - np b + 0.5 - np Z P(a X b) & P 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. McGraw-Hill/Irwin Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE BUSINESS STATISTICS 4-19 5th edi tion Using the Template for Normal Approximation of the Binomial Distribution McGraw-Hill/Irwin Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 Penutup • Sebaran Normal merupakan sebaran peluang variabel acak kontinyu yang paling banyak digunakan sebagai landasan di dalam penarikan kesimpulan/pengambilan keputusan 20