Agenda: Questions Chapter 4 so far? Review Chapter 4. Lecture Go over Homework: Chapt. #, 3 Copyright 6 by D.B. Rowe
4. Background Eample: N=5 balls in bucket, select n= with replacement. Population data values:,, 4, 6, 8. 4 6 8 P( ) / 5 / 5 4 / 5 6 / 5 8 / 5 5 possible values P()
4. Background Eample: N=5 balls in bucket, select n= with replacement. Population data values: 4 6 8 8 4 6 4 6 8,, 4, 6, 8. (,) (,) (4,) (6,) (8,) 4 6 8 4 6 8 4 6 8 4 6 8 (,) (,) (4,) (6,) (8,) (,4) (,4) (4,4) (6,4) (8,4) (,6) (,6) (4,6) (6,6) (8,6) (,8) (,8) (4,8) (6,8) (8,8) 5 possible samples 3
4. Background Eample: N=5 balls in bucket, select n= with replacement. Population data values:,, 4, 6, 8. 4 6 8 5 possible values P( ) / 5 / 5 4 / 5 6 / 5 8 / 5 [ P( )] (/ 5) (/ 5) 4(/ 5) 6(/ 5) 8(/ 5) 4 4
4. Background Eample: N=5 balls in bucket, select n= with replacement. Population data values:,, 4, 6, 8. 4 6 8 5 possible values P( ) [( ) P( )] / 5 / 5 ( 4) (/ 5) ( 4) (/ 5) 4 / 5 6 / 5 8 / 5 (4 4) (/ 5) (6 4) (/ 5) (8 4) (/ 5) 8 8 5
4. Background Eample: N=5, values:,, 4, 6, 8, n= (with replacement). P ( ) / 5 P ( ) / 5 P ( ) 3 / 5 P ( 3) 4 / 5 P ( 4) 5 / 5 P ( 5) 4 / 5 P ( 6) 3 / 5 P ( 7) / 5 P ( 8) / 5 P ( ) Represent this probability distribution with a histogram. 6
4. Background Eample: N=5, values:,, 4, 6, 8, n= (with replacement). P( ) (/ 5) ( / 5) (3 / 5) 3(4 / 5) 4(5 / 5) 5(4 / 5) 6(3 / 5) 7( / 5) 8(/ 5) 4 Same as SDSM formula! 4 P ( ) / 5 P ( ) / 5 P ( ) 3 / 5 P ( 3) 4 / 5 P ( 4) 5 / 5 P ( 5) 4 / 5 P ( 6) 3 / 5 P ( 7) / 5 P ( 8) / 5 7
4. Background Eample: N=5, values:,, 4, 6, 8, n= (with replacement). ( ) P( ) ( 4) (/ 5) ( 4) ( / 5) 4 ( 4) (3 / 5) (3 4) (4 / 5) (4 4) (5 / 5) (5 4) (4 / 5) (6 4) (3 / 5) (7 4) ( / 5) (8 4) (/ 5) Same as SDSM formula! n P ( ) / 5 P ( ) / 5 P ( ) 3 / 5 P ( 3) 4 / 5 P ( 4) 5 / 5 P ( 5) 4 / 5 P ( 6) 3 / 5 P ( 7) / 5 P ( 8) / 5 8
Chapter (Continued) Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science 9
4. The Central Limit Theorem (CLT) Assume that we have a population with mean μ and standard deviation σ. If we take random samples of size n with replacement, for large n, the distribution of the sample means is approimately normally distributed with X, X n where in general n 3 is sufficiently large.
4. The Central Limit Theorem The CLT is important because later in the course we are going to make probability statements about the population mean μ based upon a sample of data,..., n and a single sample mean. We won t know what distribution the individual observations come from, but we will know by the CLT that the mean is approimately normal, X N n ~ (, / ) X X.
4. The Central Limit Theorem Book goes through three continuous data eamples. Generate data,..., n, for n=5, 5, 3, and 5. Uniform distribution, [a=,b=]. Gamma distribution, [α=,β=] Normal distribution, [μ=,σ=] f( ) b a a b e f( ) ( ) f( ) e
4. The Central Limit Theorem Not comperable eamples. Should have same σ. Ignore second, change σ on third. Book goes through three continuous data eamples. Generate data,..., n, for n=5, 5, 3, and 5. Uniform distribution, [a=,b=]. Gamma distribution, [α=,β=] Normal distribution, [μ=,σ=] 57.7 f( ) μ =, σ =57.7 f( ) f( ) b a a b e ( ) e 3
4. The Central Limit Theorem According to CLT, if we had a sample,..., n of size n=,, 3, 4, 5, 5, 3, and 5 from Uniform distribution, [a=,b=]. Sample Size, n Mean, SD, X 57.735 4.848 3 33.3333 4 8.8675 5 5.899 5 5.899 3 4.97 5 8.65 X b b a a f( ) a b b a f ( ) d b a ( b a) ( ) f ( ) d 4
4. The Central Limit Theorem According to CLT, if we had a sample,..., n of size n=,, 3, 4, 5, 5, 3, and 5 from Normal distribution, [μ=,σ=(-)/ ] Sample Size, n Mean, SD, X 57.735 4.848 3 33.3333 4 8.8675 5 5.899 5 5.899 3 4.97 5 8.65 X f( ) f ( ) d e ( ) f ( ) d 5
4. The Central Limit Theorem I wrote a program to generate n million observations,..., n from the Uniform [a=,b=] and Normal [μ=,σ=57.7] distributions, for n=,, 3, 4, 5, 5, 3, and 5. i.e. n=5, 5 6 Groups of n=5 Mean of groups,, 3, 4, 5 6 8 data sets of Uniform and Normal... 5 6, 7 8, 9,... 4999996,.,.,., 5... 6 s Histogram of 's 6
4. The Central Limit Theorem I wrote a program to make one million means for n=,, 3, 4, 5, 5, 3, and 5. Sample Size n,..., 6 Mean Mean U Mean N SD SD U SD N X.64.77 57.735 57.77 57.7888 99.988.37 4.848 4.848 4.86 3 99.999 99.967 33.3333 33.348 33.984 4 99.9559.64 8.8675 8.8946 8.86 5.74.3 5.899 5.7865 5.8397 5.34 99.957 4.97 4.935 4.898 3 99.9934 99.9836.549.5335.535 5 99.998 99.989 8.65 8.65 8.79 X s s 7
4. The Central Limit Theorem Made Figures Analogous to Book Figures.5 5 Uniform.5 5 n=5 n=5 n=3 n=5.5 5.5 5.5.5.5.5.5.5.5.5 75 8 85 9 95 5 5 5 75 8 85 9 95 5 5 5 75 8 85 9 95 5 5 5 75 8 85 9 95 5 5 5 So what! How does this compare if data were actually from a normal population. 8
4. The Central Limit Theorem Made Figures Analogous to Book Figures.5 5 Uniform.5 5 n=5 n=5 n=3 n=5.5 5.5 5.5.5.5.5.5.5.5.5 75 8 85 9 95 5 5 5 75 8 85 9 95 5 5 5 Normal.5 5.5 5 75 8 85 9 95 5 5 5 n=5 n=5 n=3 n=5.5 5 75 8 85 9 95 5 5 5.5 5.5.5.5.5.5.5.5.5 75 8 85 9 95 5 5 5 75 8 85 9 95 5 5 5 75 8 85 9 95 5 5 5 75 8 85 9 95 5 5 5 9
4. The Central Limit Theorem Made Figures Analogous to Book Figures Uniform vs. Normal This is not the whole story. There is actually more to it!
4. The Central Limit Theorem n= 6 observations 4 9 8 7 57.73 4 9 8 7 57.73 6 5 observations are uniform 6 5 observations are normal 4 4 3 3 4 6 8 4 6 8 4 6 8 4 6 8
4. The Central Limit Theorem n= 6 observations.8.6.4 5 57.73.8.6.4 5 57.73. observations are uniform. observations are normal.8.8.6.6.4.4.. 4 6 8 4 6 8 4 6 8 4 6 8
4. The Central Limit Theorem n=3 3 6 observations.5 3 5 57.73.5 3 5 57.73.5 observations are uniform.5 observations are normal.5.5 4 6 8 4 6 8 4 6 8 4 6 8 3
4. The Central Limit Theorem n=4 4 6 observations 3.5 4 5 3 57.73 3.5 4 5 3 57.73.5 observations are uniform.5 observations are normal.5.5.5.5 4 6 8 4 6 8 4 6 8 4 6 8 4
4. The Central Limit Theorem n=5 5 6 observations 4.5 3.5 5 5 4 57.73 4.5 3.5 5 5 4 57.73 3.5 observations are uniform 3.5 observations are normal.5.5.5.5 4 6 8 4 6 8 4 6 8 4 6 8 5
4. The Central Limit Theorem n=5 5 6 observations 5 5 57.73 observations are uniform 5 5 57.73 observations are normal 5 5 4 6 8 4 6 8 4 6 8 4 6 8 6
4. The Central Limit Theorem n=3 3 6 observations.5 3 6 57.73.5 3 6 57.73.5 observations are uniform.5 observations are normal.5.5 4 6 8 4 6 8 4 6 8 4 6 8 7
4. The Central Limit Theorem n=5 5 6 observations 4.5 3.5 5 6 4 57.73 4.5 3.5 5 6 4 57.73 3.5 observations are uniform 3.5 observations are normal.5.5.5.5 4 6 8 4 6 8 4 6 8 4 6 8 8
4. The Central Limit Theorem n= 6 means 57.73.5 5.5 Histogram of means from uniform.5 5.5 Histogram of means from normal.5.5 4 6 8 4 6 8 4 6 8 4 6 8 9
4. The Central Limit Theorem n= 6 means 57.73.5 5.5 Histogram of means from uniform.5 5.5 Histogram of means from normal.5.5 4 6 8 4 6 8 4 6 8 4 6 8 3
4. The Central Limit Theorem n=3 6 means 57.73.5 5.5 Histogram of means from uniform.5 5.5 Histogram of means from normal.5.5 4 6 8 4 6 8 4 6 8 4 6 8 3
4. The Central Limit Theorem n=4 6 means 57.73.5 5.5 Histogram of means from uniform.5 5.5 Histogram of means from normal.5.5 4 6 8 4 6 8 4 6 8 4 6 8 3
4. The Central Limit Theorem n=5 6 means 57.73.5 5.5 Histogram of means from uniform.5 5.5 Histogram of means from normal.5.5 4 6 8 4 6 8 4 6 8 4 6 8 33
4. The Central Limit Theorem n=5 6 means 57.73.5 5.5 Histogram of means from uniform.5 5.5 Histogram of means from normal.5.5 4 6 8 4 6 8 4 6 8 4 6 8 34
4. The Central Limit Theorem n=3 6 means 57.73.5 5.5 Histogram of means from uniform.5 5.5 Histogram of means from normal.5.5 4 6 8 4 6 8 4 6 8 4 6 8 35
4. The Central Limit Theorem n=5 6 means 57.73.5 5.5 Histogram of means from uniform.5 5.5 Histogram of means from normal.5.5 4 6 8 4 6 8 4 6 8 4 6 8 36
4. The Central Limit Theorem (CLT) With a population mean μ and standard deviation σ. Random samples of size n with replacement, for large n, the distribution of the sample means quickly becomes normally distributed with X, X n Book says n 3 is sufficiently large but much less is fine! 37
4. The Central Limit Theorem (CLT) Eample: Applying the CLT with a μ = and σ=. Assume that we take a random sample of size n=5. What is P(99 X )? By the CLT, X, X n. 38
4. The Central Limit Theorem (CLT) Eample: For X 99: 99 Z / 5.5 For X : Z / 5 3 39
4. The Central Limit Theorem (CLT) Eample: So, P(99 X ) P(.5 Z 3) Using table B., P(.5 Z 3).9987.668.939 = - 4
Questions? Homework: Chapter 4 #, 3, 7, 9 4