Agenda: Questions Chapter 4 so far?

Transcription

1 Agenda: Questions Chapter 4 so far? Review Chapter 4. Lecture Go over Homework: Chapt. #, 3 Copyright 6 by D.B. Rowe

2 4. Background Eample: N=5 balls in bucket, select n= with replacement. Population data values:,, 4, 6, P( ) / 5 / 5 4 / 5 6 / 5 8 / 5 5 possible values P()

3 4. Background Eample: N=5 balls in bucket, select n= with replacement. Population data values: ,, 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 4. Background Eample: N=5 balls in bucket, select n= with replacement. Population data values:,, 4, 6, possible values P( ) / 5 / 5 4 / 5 6 / 5 8 / 5 [ P( )] (/ 5) (/ 5) 4(/ 5) 6(/ 5) 8(/ 5) 4 4

5 4. Background Eample: N=5 balls in bucket, select n= with replacement. Population data values:,, 4, 6, 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

6 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

7 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

8 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

9 Chapter (Continued) Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science 9

10 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.

11 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.

12 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

13 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

14 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 X b b a a f( ) a b b a f ( ) d b a ( b a) ( ) f ( ) d 4

15 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 X f( ) f ( ) d e ( ) f ( ) d 5

16 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, data sets of Uniform and Normal , 7 8, 9, ,.,.,., s Histogram of 's 6

17 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 X s s 7

18 4. The Central Limit Theorem Made Figures Analogous to Book Figures.5 5 Uniform.5 5 n=5 n=5 n=3 n= So what! How does this compare if data were actually from a normal population. 8

19 4. The Central Limit Theorem Made Figures Analogous to Book Figures.5 5 Uniform.5 5 n=5 n=5 n=3 n= Normal n=5 n=5 n=3 n=

20 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!

21 4. The Central Limit Theorem n= 6 observations observations are uniform 6 5 observations are normal

22 4. The Central Limit Theorem n= 6 observations observations are uniform. observations are normal

23 4. The Central Limit Theorem n=3 3 6 observations observations are uniform.5 observations are normal

24 4. The Central Limit Theorem n=4 4 6 observations observations are uniform.5 observations are normal

25 4. The Central Limit Theorem n=5 5 6 observations observations are uniform 3.5 observations are normal

26 4. The Central Limit Theorem n=5 5 6 observations observations are uniform observations are normal

27 4. The Central Limit Theorem n=3 3 6 observations observations are uniform.5 observations are normal

28 4. The Central Limit Theorem n=5 5 6 observations observations are uniform 3.5 observations are normal

29 4. The Central Limit Theorem n= 6 means Histogram of means from uniform Histogram of means from normal

30 4. The Central Limit Theorem n= 6 means Histogram of means from uniform Histogram of means from normal

31 4. The Central Limit Theorem n=3 6 means Histogram of means from uniform Histogram of means from normal

32 4. The Central Limit Theorem n=4 6 means Histogram of means from uniform Histogram of means from normal

33 4. The Central Limit Theorem n=5 6 means Histogram of means from uniform Histogram of means from normal

34 4. The Central Limit Theorem n=5 6 means Histogram of means from uniform Histogram of means from normal

35 4. The Central Limit Theorem n=3 6 means Histogram of means from uniform Histogram of means from normal

36 4. The Central Limit Theorem n=5 6 means Histogram of means from uniform Histogram of means from normal

37 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

38 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

39 4. The Central Limit Theorem (CLT) Eample: For X 99: 99 Z / 5.5 For X : Z /

40 4. The Central Limit Theorem (CLT) Eample: So, P(99 X ) P(.5 Z 3) Using table B., P(.5 Z 3) = - 4

41 Questions? Homework: Chapter 4 #, 3, 7, 9 4